6-1 polynomials - wikispaces6+reteach.pdf · multiplying polynomials (continued) use the...

Name ________________________________________ Date __________________ Class__________________

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6-6 Holt Algebra 2

Reteach Polynomials

The degree of a polynomial is the value of the exponent of the term of the greatest degree. A polynomial is in standard form when the terms are arranged in order with exponents from greatest to least.

To arrange the polynomial 3x 2 + x

4 − 2x + 6x 5 − 7 in standard form, order the terms from

greatest to least exponent. 6x

5 + x 4 + 3x

2 − 2x − 7

Rewrite each polynomial in standard form. Then identify the leading coefficient, degree, and number of terms of each polynomial. 1. 2x + x

3 − x 2 − 5 2. 5x

2 + 3x 4 − x

Standard form: x 3 − x

2 + 2x − 5 Standard form: 3x 4 + ____________

Leading coefficient: 1 Leading coefficient: ____________

Degree: _______________ Degree: _____________

Number of terms: _______________ Number of terms: ______________

3. 6x 3 + 7x

5 4. −3x 2 + x

4 − x − 2x 3 + 8

Standard form: ______________ Standard form: _________________

Leading coefficient: _______________ Leading coefficient: _____________

Degree: ________________ Degree: _____________

Number of terms: ________________ Number of terms: _______________

LESSON

6-1

Degree Polynomial in Standard Form 0 8 1 2x + 3 2 −x

2 + 4x − 5 3 4x

3 − x 4 6x

4 + x 3 − 5x

2 + 3x − 1 5 9x

5 + x 3 − 1

Constants have degree 0.

This third degree polynomial has 2 terms.

This fifth degree polynomial has 3 terms.

6 is the leading coefficient of this polynomial.

Name ________________________________________ Date __________________ Class__________________


6-7 Holt Algebra 2

Reteach Polynomials (continued)

To add polynomials: − Write each polynomial in standard form. − Align like terms vertically. − Add like terms.

Add: ( ) ( )3 2 2 36 2 5 1 4 2 .x x x x x x+ − − + + +

2x 3 − 5x

2 + 6x − 1 + x

3 + 4x 2 + 2x

3x 3

− x 2 + 8x − 1

To subtract polynomials, add the opposite vertically.

Subtract: ( ) ( )3 2 2 36 2 5 1 4 2 .x x x x x x+ − − − + +

Add the opposite: ( ) ( )3 2 2 36 2 5 1 4 2 .x x x x x x+ − − + − − −

2x 3 − 5x

2 + 6x − 1 + (−x

3 − 4x 2 − 2x)

x 3 − 9x

2 + 4x − 1

Write each polynomial in standard form. Add or subtract.

5. ( ) ( )2 3 23 2 6 2 1x x x x x+ − + + + 6. ( ) ( )3 2 34 5 4 2x x x x x+ − + − +

2x 3 + 3x

2 − x ______________________ + 2x

2 + 6x + 1 + ______________________

_________________________________________ ________________________________________

7. ( ) ( )2 26 4 1 2 1x x x x+ − − − + 8. ( ) ( )3 2 34 6 3x x x+ − −

Add the opposite: Add the opposite:

( ) ( )2 36 4 1 2 1x x x x+ − + − + − _________________________________

6x 2 + 4x − 1 _________________________________

+ x 3 − 2x − 1 + _______________________

_________________________________________ ________________________________________

LESSON

6-1

Standard form: x 3 + 4x

2 + 2x

Standard form: 2x 3 − 5x

2 + 6x − 1

Align like terms.

Add like terms vertically.

Add like terms vertically.

Name ________________________________________ Date __________________ Class__________________


6-14 Holt Algebra 2

Reteach Multiplying Polynomials

Use the Distributive Property to multiply a monomial and a polynomial. Think: k(x + y + z) = kx + ky + kz Multiply: 2ab

2(3a 2b − 4ab

2 − b 3).

2ab 2(3a

2b − 4ab 2 − b

3) 2ab

2(3b 2b) + 2ab

2(−4ab 2) + 2ab

2(−b 3) Distribute 2ab

2.

2(3)(a ⋅ a 2)(b

2 ⋅ b) + 2(−4)(a • a)(b 2 ⋅ b

2) + 2(−1)(a)(b 2 ⋅ b

3) Group like terms.

6a 3b

3 − 8a 2b

4 − 2ab 5 Multiply.

Find each product. 1. 4x

2(x 2 + 2x − 3) 2. c

2d 2(3c

2 − cd + 7d 2)

4x 2(x

2) + 4x 2(2x) + 4x

2(−3) c 2d

2(3c 2) + c

2d 2(−cd) + c

2d 2(7d

2)

4x 2(x

2) + 4(2)(x 2 ⋅ x) + 4(−3)x

2 3(c 2 ⋅ c

2)(d 2) − (c

2 ⋅ c)(d 2 ⋅ d)

+ 7c 2(d

2 • d 2)

_________________________________________ ________________________________________

3. 5xy 2(x

3 + 4x 2 + 2) 4. 3a

2b 2(8a

2 − 2ab − b 2)

5xy 2(x

3) + 4xy 2(4x

2) + 5xy 2(2) 3a

2b 2(8a

2) + 3a 2b

2(−2ab) + 3a

2b 2(−b

2)

_________________________________________ ________________________________________

_________________________________________ ________________________________________

5. 2y 3(y

2 − 9y + 4) 6. x 2y

2(4x 2 + 7y)

_________________________________________ ________________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

LESSON

6-2

2ab 2 is a monomial. 3a

2b − 4ab 2 − b

3 is a polynomial.

Remember: Add the exponents of like bases to multiply.

Name ________________________________________ Date __________________ Class__________________


6-15 Holt Algebra 2

Reteach Multiplying Polynomials (continued)

Use the Distributive Property to multiply two polynomials. Distribute each term of the first polynomial to each term of the second polynomial. Multiply: (x + 2)(4x

2 − 3x − 1). Horizontal Method: (x + 2)(4x

2 − 3x − 1)

[x(4x 2) + x(−3x) + x(−1)] + [2(4x

2) + 2(−3x) + 2(−1)] 4x

3 − 3x 2 − x + 8x

2 − 6x − 2 Multiply. 4x

3 − 3x 2 + 8x

2 − x − 6x − 2 Group like terms. 4x

3 + 5x 2 − 7x − 2 Combine like terms.

Vertical Method: 4x 2 − 3x − 1

x + 2 8x

2 − 6x − 2 4x

3 − 3x 2 − x

4x 3 + 5x

2 − 7x − 2

Use the horizontal method to find each product.

7. (x − 3)(x 2 − 2x + 2) 8. (a + b)(a

2 + ab − 4b)

x(x 2) + x(−2x) + x(2) − 3(x

2) − 3(−2x) −3(2) _________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

Use the vertical method to find each product.

9. x 2 + 4x − 6 10. y

2 − 3y + 1 x + 2 y − 1

_________________________________________ ________________________________________

LESSON

6-2

Align like terms.

Combine like terms.

Multiply 4x 2 − 3x − 1 by 1.

Multiply 4x 2 − 3x − 1 by x.

Distribute x to each term of (4x

2 − 3x − 1). Distribute 2 to each termof (4x

2 − 3x − 1).

Name ________________________________________ Date __________________ Class__________________


6-22 Holt Algebra 2

Reteach Dividing Polynomials

In arithmetic long division, you follow these steps: divide, multiply, subtract, and bring down. Follow these same steps to use long division to divide polynomials. Divide: (6x

2 + x + 8) ÷ (2x − 1). Step 1 Divide the first term of the dividend, 6x

2, by the first term of the divisor, 2x.

2

32 1 6 8

xx x x− + + Divide: 6x

2 ÷ 2x = 3x.

− (6x 2 − 3x) Multiply the complete divisor: 3x(2x − 1) = 6x

2 − 3x. 4x + 8 Subtract and bring down. Step 2 Divide the first term of the difference, 4x, by the first term of the divisor, 2x.

+

− + +2

3 22 1 6 8

xx x x

− (6x 2 − 3x) Multiply: 3x(2x − 1) = 6x

2 − 3x. 4x + 8 Divide: 4x ÷ 2x = 2. − (4x − 2) Multiply the complete divisor: 2(2x − 1) = 4x − 2. 10 Subtract. Use the Distributive Property. Step 3 Write the quotient including the remainder.

( ) ( )+ + ÷ − = + +−

2 106 8 2 1 3 22 1

x x x xx

Use long division to divide.

1.

( )

2

2

42 4 7 6

4 8

6

xx x x

x x

x

+ + +

− +

− +

2. + + +24 2 9 9x x x

_________________________________________ ________________________________________

3. − − −25 3 5 50x x x 4. + + −23 2 6 7 6x x x

_________________________________________ ________________________________________

LESSON

6-3

Remember to use the Distributive Property when you subtract.

Name ________________________________________ Date __________________ Class__________________


6-23 Holt Algebra 2

Reteach Dividing Polynomials (continued)

When the divisor is in the form (x − a), use synthetic division to divide. Divide: (2x

2 − x − 10) ÷ (x − 3). Step 1 Find a. The divisor is (x − 3). So, a = 3. Step 2 Write a in the upper left corner. Then write the coefficients of the dividend. − −3 2 1 10

Step 3 Draw a horizontal line. Copy the first coefficient below the line.

− −3 2 1 10

2

Step 4 Multiply the first coefficient by a, or 3. Write the product in the second column. Add the numbers in the column.

− −3 2 1 106

2 5

Step 5 Multiply that sum by a, or 3. Write the product in the third column. Add the numbers in the column. Draw a box around the last number. It is the remainder.

− −3 2 1 106 15

2 5 5

Step 6 Write the quotient. + +−52 5

3x

x

Use synthetic division to divide. 5. (4x

2 + 7x + 10) ÷ (x + 2) 6. (2x 2 − 6x − 12) ÷ (x − 5)

a = −2 a = ______________

−

−2 4 7 10

84

− −2 6 12

_________________________________________ ________________________________________

LESSON

6-3

2, −1, and −10 are the coefficients of 2x 2 − x − 10.

2a = 2(3) = 6

5a = 5(3) = 15

The numbers in the bottom row are the coefficients of the quotient.

Name ________________________________________ Date __________________ Class__________________


6-30 Holt Algebra 2

Reteach Factoring Polynomials

Sometimes you can use grouping to factor a third degree polynomial. To factor by grouping means to group terms with common factors. Then factor the common factors. Continue to factor until the expression can no longer be factored. Factor: x

3 + 4x 2 − 9x − 36.

Start by grouping terms to factor out the greatest possible power of x.

x 3 + 4x

2 − 9x − 36

(x 3 + 4x

2) + (−9x − 36)

x 2(x + 4) − 9(x + 4)

(x + 4) (x 2 − 9)

(x + 4) (x + 3) (x − 3)

Factor each expression. 1. x

3 − 3x 2 − 4x + 12 2. x

3 + 6x 2 − x − 6

(x 3 − 3x

2) + (−4x + 12) (x 3 + 6x

2) + (−x − 6)

x 2(x − 3) − 4(x − 3) ________________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

3. x 3 + x

2 − 9x − 9 4. x 3 + 2x

2 − 16x − 32

_________________________________________ ________________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

LESSON

6-4

−9 is a factor of −9 and −36.x 2 is a factor of

x and 4x 2.

(x + 4) is a common factor.

(x 2 − 9) is the difference of squares.

Recall that (a 2 − b

2) = (a + b)(a − b). So (x

2 − 9) = (x + 3) (x − 3).

Name ________________________________________ Date __________________ Class__________________


6-31 Holt Algebra 2

Reteach Factoring Polynomials (continued)

Use special rules to factor the sum or difference of two cubes. Recognizing these common cubes can help you factor the sum or difference of cubes. 1

3 = 1, 2 3 = 8, 3

3 = 27, 4 3 = 64, 5

3 = 125, and 6 3 = 216

Rule for the Sum of Two Cubes: a 3 + b

3 = (a + b)(a 2 − ab + b

2). Factor: y

3 + 64.

y 3 + 64 Identify the cubes: y

3 and 64 = 4 3.

y 3 + 4

3 Write the expression as the sum of two cubes.

(y + 4)(y 2 − 4y + 16) Use the rule to factor.

Rule for the Difference of Two Cubes: a 3 − b

3 = (a − b) (a 2 + ab + b

2). Factor: 8x

3 − 125. 8x

3 − 125 Identify the cubes: (2x) 3 and 125 = 5

3. (2x)

3 − 5 3 Write the expression as the difference of two cubes.

(2x − 5)(4x 2 + 10x + 25) Use the rule to factor.

Factor each expression. 5. 27x

3 + 8 6. y 3 − 216

(3x) 3 + 2

3 y 3 − 6

3

_________________________________________ ________________________________________

7. y 3 + 27 8. x

3 − 1

_________________________________________ ________________________________________

_________________________________________ ________________________________________

LESSON

6-4

Using the rule: a = y and b = 4. So a

2 = y 2, ab = 4y, and b

2 = 16.

Using the rule: a = 2x and b = 5. So a

2 = (2x) 2 = 4x

2, ab = (2x)(5) = 10x, and b 2 = 25.

Name ________________________________________ Date __________________ Class__________________


6-38 Holt Algebra 2

Reteach Finding Real Roots of Polynomial Equations

To find the roots of a polynomial equation, set the equation equal to zero. Factor the polynomial expression completely. Then set each factor equal to zero to solve for the variable. Solve the equation: 2x

5 + 6x 4 = 8x

3. Step 1 To set the equation equal to 0, rearrange the equation so that all the terms are on

one side. 2x

5 + 6x 4 = 8x

3 2x

5 + 6x 4 − 8x

3 = 0 Step 2 Look for the greatest number and the greatest power of x that can be factored from

each term. 2x

5 + 6x 4 − 8x

3 = 0 2x

3(x 2 + 3x − 4) = 0

Step 3 Factor the quadratic. 2x

3(x 2 + 3x − 4) = 0

2x 3(x + 4) (x − 1) = 0

Step 4 Set each factor equal to 0. 2x

3 = 0 x + 4 = 0 x − 1 = 0 Step 5 Solve each equation. 2x

3 = 0 x + 4 = 0 x − 1 = 0 x = 0 x = −4 x = 1 The solutions of the equation are called the roots. The roots are −4, 0, and 1.

Solve each polynomial equation. 1. 3x

6 − 9x 5 = 30x

4 2. x 4 + 6x

2 = 5x 3

3x 6 − 9x

5 − 30x 4 = 0 x

4 − 5x 3 + 6x

2 = 0 3x

4(x 2 − 3x − 10) = 0

_________________________________________ ________________________________________

_________________________________________ ________________________________________

3. 2x 3 − 6x

2 − 36x = 0 4. 2x 6 − 32x

4 = 0

_________________________________________ ________________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

LESSON

6-5

The GCF is 2x 3.

Name ________________________________________ Date __________________ Class__________________


6-39 Holt Algebra 2

Reteach Finding Real Roots of Polynomial Equations (continued)

You can use the Rational Root Theorem to find rational roots. Use the Rational Root Theorem. Solve the equation: x

3 + 3x 2 − 6x − 8 = 0.

The constant term is −8. The leading coefficient is 1. p: factors of −8 are ±1, ±2, ±4, ±8 q: factors of 1 are ±1

Possible roots, pq

: ±1, ±2, ±4, ±8

Test some possible roots to find an actual root. Use a synthetic substitution table. The first column lists possible roots. The last column represents the remainders. A root has a remainder of 0. 2 is a root, so x − 2 is a factor. Use the coefficients from the table to write the other factor. (x − 2) (x

2 + 5x + 4) = 0 (x − 2) (x + 4) (x + 1) = 0 x = 2 or x = −4 or x = −1 The roots of the equation are −4, −1, and 2.

Use the Rational Root Theorem. Solve x 3 − 7x

2 + 7x + 15 = 0. 5. a. Identify possible roots. ____________________________ b. Use the synthetic substitution table to identify an actual root.

c. Write the factors of the equation.

_________________________________________________________________________________________ d. Identify the roots of the equation.

_________________________________________________________________________________________

LESSON

6-5

Rational Root Theorem If a polynomial has integer coefficients, then every rational root

can be written in the form pq

, where p is a factor of the constant

term and q is a factor of the leading coefficient.

Coefficients of the Equationpq

1 3 −6 −8

1 1 4 −2 −10

2 1 5 4 0

4 1 7 22 80

Factor the quadratic to find the other factors.

Coefficients of the Equationpq

1 −7 7 15

Name ________________________________________ Date __________________ Class__________________


6-46 Holt Algebra 2

Reteach Fundamental Theorem of Algebra

If r is a root of a polynomial function, then (x − r) is a factor of the polynomial, P(x) . So, you can use the roots to write the simplest form of a polynomial function. Write the simplest polynomial function with roots −4, −2, and 3. Step 1 Write the factors of the polynomial, P(x) = 0. (x + 4)(x + 2)(x − 3) = 0 Step 2 Multiply the first two factors, (x + 4)(x + 2). (x

2 + 6x + 8)(x − 3) = 0 Step 3 Multiply (x

2 + 6x + 8)(x − 3). Then simplify. x

3 − 3x 2 + 6x

2 − 18x + 8x − 24 = 0 x

3 + 3x 2 − 10x − 24 = 0

The function is P(x) = x 3 + 3x

2 − 10x − 24 = 0.

Write the simplest polynomial function with the given roots.

1. −5, 1, and 2 2. −3, −1, and 0

(x + 5)(x − 1)(x − 2) = 0 x(x + 3)(x + 1) = 0

(____________)(x − 2) = 0

_________________________________________ ________________________________________

_________________________________________ ________________________________________

3. 1, 4, and 5 4. −2, 3, and 6

_________________________________________ ________________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

5. 2, 4, and 6 6. −5, 0, and 5

_________________________________________ ________________________________________

_________________________________________ ________________________________________

_________________________________________ ________________________________________

LESSON

6-6

Root (a) −4 −2 3

Factor (x − a)

x + 4 x + 2 x − 3

Name ________________________________________ Date __________________ Class__________________


6-47 Holt Algebra 2

Reteach Fundamental Theorem of Algebra (continued)

To solve x 4 + x

3 − 5x 2 + x − 6 = 0 means to find all the roots

of the equation. A fourth degree equation has 4 roots. Step 1 Identify possible real roots.

Possible roots, pq

: ±1, ±2, ±3, ±6

Step 2 Graph y = x 4 + x

3 − 5x 2 + x − 6.

Step 3 Test 2 as a root using synthetic substitution.

2 1 1 5 1 6

2 6 2 61 3 1 3 0

− −

Test −3 as a root using synthetic substitution.

−

− −3 1 3 1 3

3 0 31 0 1 0

Step 4 Find the remaining roots. x

2 + 1 = 0 x = ± i The roots of the equation are 2, −3, i, and −i.

Find the roots of the equation x 4 − 3x

3 + 6x 2 − 12x + 8 = 0.

7. Possible roots: ±1, ±2, ±4, ±8 Test: _____________ and _____________. _____________ and _____________ are real roots. Solve _____________ to find the remaining roots. Remaining roots: _____________ and _____________.

LESSON

6-6

The factors, p, of −6 are ±1, ±2, ±3, ±6. The factors, q, of 1 are ±1.

The remainder is 0, so 2 is a root. (x − 2)(x

3 + 3x 2 + x + 3) = 0

The remainder is 0, so −3 is a root. (x − 2) (x + 3) (x

2 + 1) = 0

Name ________________________________________ Date __________________ Class__________________


6-54 Holt Algebra 2

Reteach Investigating Graphs of Polynomial Functions

Examine the sign and the exponent of the leading term (term of greatest degree) of a polynomial P(x) to determine the end behavior of the function. Even degree functions: Exponent of leading term is even. Example: P(x) = 3x

4 + 2x 3 − 5 Leading term: 3x

4 End behavior: As x → −∞, P(x) → +∞.

As x → +∞, P(x) → +∞. Odd degree functions: Exponent of leading term is odd. Example: P(x) = −2x

5 − 6x 2 + x Leading term: −2x

5 End behavior: As x → −∞, P(x) → +∞.

As x → +∞, P(x) → −∞.

Identify the end behavior of each function. 1. P(x) = 4x

3 + 8x 2 − 5 2. P(x) = −9x

6 + 2x 3 − x + 7

Leading term: 4x 3 Leading term: _______________

Sign and degree: ________________ Sign and degree: ______________ End behavior: ___________________ End behavior:___________________

_________________________________________ ________________________________________

Positive leading coefficient As x → −∞, P(x) → +∞. As x → +∞, P(x) → +∞.

Negative leading coefficientAs x → −∞, P(x) → −∞. As x → +∞, P(x) → −∞.

Read: As x approaches

positive infinity, P(x) approaches negative

infinity.

Sign: positive Degree: 4, even

Positive leading coefficient As x → +∞, P(x) → +∞. As x → −∞, P(x) → −∞.

Negative leading coefficient As x → −∞, P(x) → +∞. As x → +∞, P(x) → −∞.

Sign: negative Degree: 5, odd

LESSON

6-7

Name ________________________________________ Date __________________ Class__________________


6-55 Holt Algebra 2

Reteach Investigating Graphs of Polynomial Functions (continued)

You can use the graph of a polynomial function to analyze the function.

Identify whether each function has an odd or even degree and a positive or negative leading coefficient. 3. 4. 5.

________________________ _________________________ ________________________

LESSON

6-7

P(x): odd degree and positive leading coefficient if: As x → −∞, P(x) → −∞ and as x → +∞, P(x) x → +∞.

P(x): odd degree and negative leading coefficient if: As x → −∞, P(x) x → +∞ and as x → +∞, P(x) x → −∞.

P(x): even degree and positive leading coefficient if: As x → −∞, P(x) x → +∞ and as x → +∞, P(x) x → +∞.

P(x): even degree and negative leading coefficient if: As x → −∞, P(x) → −∞ and as x → +∞, P(x) → −∞.

Notice the graph increases, then decreases, and then increases again.

Notice this graph is the reverse. It decreases, then increases, and then decreases again.

Look at the end behavior. The graph increases at both ends.

This is the reverse. The graph decreases at both ends.

Name ________________________________________ Date __________________ Class__________________


6-62 Holt Algebra 2

Reteach Transforming Polynomial Functions

Translations of polynomial functions shift the graph of the function right, left, up, or down.

For f (x) = x 3 + 2, write the rule for each function and sketch its graph.

1. g(x) = f (x) + 1 2. g(x) = f (x − 3) Translate f (x) 1 unit ______________. Translate f (x) 3 units ______________. g(x) = _____________________ g(x) = _____________________

LESSON

6-8

Vertical Translation

If f (x) is a polynomial function, g(x) = f (x) + k is a vertical translation of f (x). Example: f (x) = x

3 + 2

Think: Add to y, go high. f (x) shifts up for k > 0. f (x) shifts down for k < 0.

Vertical translation 5 units down g(x) = f (x) − 5 g(x) = x

3 + 2 − 5 g(x) = x

3 − 3

Horizontal Translation

If f(x) is a polynomial function, g(x) = f (x − h) is a horizontal translation of f (x). Example: f (x) = x

3 + 2

Think: Add to x, go west. f (x) shifts right for h > 0. f (x) shifts left for h < 0.

Horizontal translation 4 units left

g(x) = ( )( 4)f x − −

g(x) = (x + 4) 3 + 2

To graph g(x), move the graph of f (x) 5 units down.

To graph g(x), move the graph of f (x) 4 units left.

Name ________________________________________ Date __________________ Class__________________


6-63 Holt Algebra 2

Reteach Transforming Polynomial Functions (continued)

Stretches and compressions are transformations of polynomial functions.

Let f (x) = 2x 4 − 6x

2 + 4. Describe g(x) as a transformation of f (x) and write the rule for g(x). 3. g(x) = 2f (x) 4. g(x) = f (2x)

_________________________________________ ________________________________________

_________________________________________ ________________________________________

LESSON

6-8

Vertical Stretch or Compression

If f (x) is a polynomial function, g(x) = af (x) is a vertical stretch or compression of f (x). Example: f (x) = 2x

4 − 6x 2 + 4

Vertical stretch if a > 1 Vertical compression if 0 < a < 1

Vertical compression of f (x) 1( ) ( )2

g x f x=

4 21( ) (2 6 4)2

g x x x= − +

g(x) = x 4 − 3x

2 + 2

Horizontal Stretch or Compression

If f (x) is a polynomial function, 1( )g x f xb

⎛ ⎞= ⎜ ⎟⎝ ⎠

is a horizontal stretch

or compression of f (x). Example: f (x) = 2x

4 − 6x 2 + 4

Horizontal stretch if b > 1 Horizontal compression if 0 < b < 1

Horizontal stretch of f(x)

1( )2

g x f x⎛ ⎞= ⎜ ⎟⎝ ⎠

4 21 1( ) 2 6 42 2

g x x x⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

4 21 3( ) 48 2

g x x x= − +

g(x) = 2x 4 − 6x

2 + 4

g(x) = x 4 − 3x

2 + 2

g(x) = 2x 4 − 6x

2 + 4

4 21 3( ) 48 2

g x x x= − +

Name ________________________________________ Date __________________ Class__________________


6-70 Holt Algebra 2

Reteach Curve Fitting with Polynomial Models

To use finite differences to determine the degree of a polynomial, − check that the x-values increase by a constant value, and − find successive differences of the y-values until the differences are constant. Example: A fourth degree polynomial best describes the data.

Use finite differences to determine the degree of the polynomial that best describes the data. 1.

x −2 −1 0 1 2 y −5 2 3 4 11 First

Differences

Second Differences

Third Differences

2. Identify the degree of the polynomial.

LESSON

6-9

Finite Differences

Function Type Linear Quadratic Cubic Quartic Quintic

Degree 1 2 3 4 5

Constant Finite Differences First Second Third Fourth Fifth

x −3 −2 −1 0 1 2

y 78 14 0 0 2 18

14 − 78 0 − 14 0 − 0 2 − 0 18 − 2 First Differences

−64 −14 0 2 16

−14 − (−64) 0 − (−14) 2 − 0 16 − 2 Second Differences

50 14 2 14

14 − 50 2 − 14 14 − 2 Third Differences

−36 −12 12

−12 − (−36) 12 − (−12) Fourth Differences

24 24

The x-values increase by 1.

First differences are not constant.

Second differences are not constant.

Third differences are not constant.

Fourth differences are constant.

Name ________________________________________ Date __________________ Class__________________


6-71 Holt Algebra 2

Reteach Curve Fitting with Polynomial Models (continued)

Use finite differences that are close to select a polynomial model to fit a data set. Then use your calculator to write the function. Since the third differences are reasonably close, you can use a cubic function to model the data. Use the cubic regression feature on your calculator. Use the coefficients a, b, c, and d to write the function. f(x) ≈ 0.12x

3 − 8.06x 2 + 273.1x + 584.6

Write a polynomial function for the data. 3.

x 2 4 6 8 10 12

y −12 −15 38 190 446 773

First Differences

Second Differences

Third Differences

Fourth Differences

4. Write a polynomial function that best describes the data set.

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x 10 20 30 40 50

y 2633 3812 4862 6529 9552

3812 − 2633 4862 − 3812 6529 − 4862 9552 − 6529 First Differences

1179 1050 1667 3023

1050 − 1179 1667 − 1050 3023 − 1667 Second Differences

−129 617 1356

617 − (−129) 1356 − 617 Third Differences

746 739

LESSON

6-9

6-1 polynomials - wikispaces6+reteach.pdf · multiplying polynomials (continued) use the...

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