10-6 dividing polynomials warm up warm up lesson presentation lesson presentation california...
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10-6 Dividing Polynomials
Warm UpDivide.
1. m2n ÷ mn4 2. 2x3y2 ÷ 6xy
3. (3a + 6a2) ÷ 3a2b
Factor each expression.
4. 5x2 + 16x + 12
5. 16p2 – 72p + 81
10-6 Dividing Polynomials
California Standards
10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, by using these techniques.
12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.
10-6 Dividing Polynomials
To divide a polynomial by a monomial, you can first write the division as a rational expression. Then divide each term in the polynomial by the monomial.
10-6 Dividing PolynomialsAdditional Example 1: Dividing a Polynomial by a
Monomial
Divide (5x3 – 20x2 + 30x) ÷ 5x
x2 – 4x + 6
Write as a rational expression.
Divide each term in the polynomial by the monomial 5x.
Divide out common factors.
Simplify.
10-6 Dividing Polynomials
Check It Out! Example 1a Divide.
(8p3 – 4p2 + 12p) ÷ (–4p2)
Write as a rational expression.
Divide each term in the polynomial by the monomial –4p2.
Divide out common factors.
Simplify.
10-6 Dividing Polynomials
Check It Out! Example 1b
Divide.
(6x3 + 2x – 15) ÷ 6x
Write as a rational expression.
Divide each term in the polynomial by the monomial 6x.
Divide out common factors in each term.
Simplify.
10-6 Dividing Polynomials
Division of a polynomial by a binomial is similar to division of whole numbers.
10-6 Dividing Polynomials
Additional Example 2A: Dividing a Polynomial by a Binomial
Divide.
x + 5
Factor the numerator.
Divide out common factors.
Simplify.
10-6 Dividing PolynomialsAdditional Example 2B: Dividing a Polynomial by a
BinomialDivide.
Factor both the numerator and denominator.
Divide out common factors.
Simplify.
10-6 Dividing Polynomials
Put each term of the numerator over the denominator only when the denominator is a monomial. If the denominator is a polynomial, try to factor first.
Helpful Hint
10-6 Dividing Polynomials
Check It Out! Example 2a
Divide.
k + 5
Factor the numerator.
Divide out common factors.
Simplify.
10-6 Dividing PolynomialsCheck It Out! Example 2b
Divide.
b – 7
Factor the numerator.
Divide out common factors.
Simplify.
10-6 Dividing Polynomials
Check It Out! Example 2c
Divide.
s + 6
Factor the numerator.
Divide out common factors.
Simplify.
10-6 Dividing Polynomials
Recall how you used long division to divide whole numbers as shown at right. You can also use long division to divide polynomials. An example is shown below.
) x2 + 3x + 2x + 1
x2 + 2xx + 2x + 2
0
x + 2
(x2 + 3x + 2) ÷ (x + 2)
Divisor Quotient
Dividend
10-6 Dividing Polynomials
Using Long Division to Divide a Polynomial by a Binomial
Step 1 Write the binomial and polynomial in standard form.
Step 3 Multiply this first term of the quotient by the binomial divisor and place the product under the dividend, aligning like terms.
Step 2 Divide the first term of the dividend by the first term of the divisor. This the first term of the quotient.
Step 4 Subtract the product from the dividend.
Step 5 Bring down the next term in the dividend.
Step 6 Repeat Steps 2-5 as necessary until you get 0 or until the degree of the remainder is less than the degree of the binomial.
10-6 Dividing PolynomialsAdditional Example 3A: Polynomial Long Division
Divide using long division. Check your answer.
(x2 +10x + 21) ÷ (x + 3)
x2 + 10x + 21)Step 1 x + 3Write in long division form
with expressions in standard form.
Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
x2 + 10x + 21)Step 2 x + 3x
10-6 Dividing PolynomialsAdditional Example 3A Continued
Divide using long division.
(x2 +10x + 21) ÷ (x + 3)
Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms.
x2 + 10x + 21)Step 3 x + 3x
x2 + 3x
x2 + 10x + 21)Step 4 x + 3–(x2 + 3x)
x
0 + 7x
Subtract the product from the dividend.
10-6 Dividing Polynomials
Additional Example 3A Continued
Divide using long division.
x2 + 10x + 21)Step 5 x + 3–(x2 + 3x)
x
+ 21
Bring down the next term in the dividend.
Repeat Steps 2-5 as necessary.
x2 + 10x + 21)Step 6 x + 3–(x2 + 3x)
x + 7
7x + 21–(7x + 21)
0The remainder is 0.
7x
10-6 Dividing Polynomials
Additional Example 3A Continued
Check: Multiply the answer and the divisor.
(x + 3)(x + 7)
x2 + 7x + 3x + 21
x2 + 10x + 21
10-6 Dividing Polynomials
When the remainder is 0, you can check your simplified answer by multiplying it by the divisor. You should get the numerator.
Helpful Hint
10-6 Dividing PolynomialsAdditional Example 3B: Polynomial Long Division
Divide using long division.
x2 – 2x – 8 )x – 4 Write in long division form.
–(x2 – 4x)2x
x2 – 2x – 8)x – 4
–(2x – 8)
0
x2 ÷ x = xMultiply x (x – 4). Subtract.
Bring down the 8. 2x ÷ x = 2.
Multiply 2(x – 4). Subtract.The remainder is 0.
x+ 2
– 8
10-6 Dividing Polynomials
Additional Example 3B Continued
Check: Multiply the answer and the divisor.
(x + 2)(x – 4)
x2 – 4x + 2x – 8
x2 – 2x + 8
10-6 Dividing PolynomialsCheck It Out! Example 3a
Divide using long division.
(2y2 – 5y – 3) ÷ (y – 3)
2y2 – 5y – 3 )Step 1 y – 3Write in long division form
with expressions in standard form.
Divide the first term of the dividend by the first term of the divisor to get the first term of the quotient.
2y2 – 5y – 3)Step 2 y – 32y
10-6 Dividing Polynomials
Check It Out! Example 3a Continued
Divide using long division.
(2y2 – 5y – 3) ÷ (y – 3)
Multiply the first term of the quotient by the binomial divisor. Place the product under the dividend, aligning like terms.
Subtract the product from the dividend.
2y2 – 5y – 3)Step 3 y – 32y
2y2 – 6y
–(2y2 – 6y)0 + y
2y2 – 5y – 3)Step 4 y – 3
2y
10-6 Dividing Polynomials
Check It Out! Example 3a Continued
Divide using long division.
)Step 5 y – 3
2y
– 3
Bring down the next term in the dividend.
Repeat Steps 2–5 as necessary.
2y
2y2 – 5y – 3 )Step 6 y – 3–(2y2 – 6y)
y – 3 –(y – 3)
0The remainder is 0.
2y2 – 5y – 3–(2y2 – 6y)
y
+ 1
10-6 Dividing Polynomials
Check: Multiply the answer and the divisor.
(y – 3)(2y + 1)
2y2 + y – 6y – 3
2y2 – 5y – 3
Check It Out! Example 3a Continued
10-6 Dividing PolynomialsCheck It Out! Example 3b
Divide using long division.
(a2 – 8a + 12) ÷ (a – 6)
a2 – 8a + 12 )a – 6 Write in long division form.
–(a2 – 6a)–2a
a a2 – 8a + 12)a – 6
–(–2a + 12)
0
a2 ÷ a = a
Multiply a (a – 6). Subtract.
Bring down the 12. –2a ÷ a = –2.
Multiply –2(a – 6). Subtract.
The remainder is 0.
– 2
+ 12
10-6 Dividing Polynomials
Check It Out! Example 3b Continued
Check: Multiply the answer and the divisor.
(a – 6)(a – 2)
a2 – 2a – 6a + 12
a2 – 8a + 12
10-6 Dividing Polynomials
Sometimes the divisor is not a factor of the dividend, so the remainder is not 0. Then the remainder can be written as a rational expression.
10-6 Dividing Polynomials
Additional Example 4: Long Division with a Remainder
Divide (3x2 + 19x + 26) ÷ (x + 5)
3x2 + 19x + 26 )x + 5 Write in long division form.
3x2 + 19x + 26 )x + 53x
–(3x2 + 15x)4x
3x2 ÷ x = 3x.Multiply 3x(x + 5). Subtract.
Bring down the 26. 4x ÷ x = 4.
Multiply 4(x + 5). Subtract.–(4x + 20)
6 The remainder is 6.
Write the remainder as a rational expression using the divisor as the denominator.
+ 4
+ 26
10-6 Dividing Polynomials
Additional Example 4 Continued
Divide (3x2 + 19x + 26) ÷ (x + 5)
Write the quotient with the remainder.
10-6 Dividing Polynomials
Check It Out! Example 4a
Divide.
3m2 + 4m – 2 )m + 3 Write in long division form.
3m2 + 4m – 2 )m + 33m
–(3m2 + 9m)
3m2 ÷ m = 3m.Multiply 3m(m + 3). Subtract.
Bring down the –2. –5m ÷ m = –5 .
Multiply –5(m + 3). Subtract.–5m
The remainder is 13.13
–(–5m – 15)
– 5
– 2
10-6 Dividing Polynomials
Check It Out! Example 4a Continued
Divide.
Write the remainder as a rational expression using the divisor as the denominator.
10-6 Dividing PolynomialsCheck It Out! Example 4b
Divide.
y2 + 3y + 2 )y – 3 Write in long division form.
–(y2 – 3y)
y2 ÷ y = y.Multiply y(y – 3). Subtract.
Bring down the 2. 6y ÷ y = 6.
y y2 + 3y + 2 )y – 3
Multiply 6(y – 3). Subtract.
The remainder is 20.20
6y –(6y –18)
Write the quotient with the remainder.
+ 6
+ 2
y + 6 +
10-6 Dividing Polynomials
Sometimes you need to write a placeholder for a term using a zero coefficient. This is best seen if you write the polynomials in standard form.
10-6 Dividing Polynomials
Additional Example 5: Dividing Polynomials That Have a Zero Coefficient
Divide (x3 – 7 – 4x) ÷ (x – 3).
x3 + 0x2 – 4x – 7 )x – 3 x3 ÷ x = x2
Multiply x2(x – 3). Subtract.
(x3 – 4x – 7) ÷ (x – 3) Write the polynomials in standard form.
Write in long division form. Use 0x2 as a placeholder for the x2 term. x2
x3 + 0x2 – 4x – 7 )x – 3
–(x3 – 3x2)
3x2 – 4x Bring down –4x.
10-6 Dividing Polynomials
Additional Example 5 Continued
x3 + 0x2 – 4x – 7 )x – 3 3x3 ÷ x = 3xMultiply x2(x – 3). Subtract.
x2
–(x3 – 3x2)
3x2 – 4x Bring down –4x.–(3x2 – 9x)
5x–(5x – 15)
8
Bring down – 7.
Multiply 3x(x – 3). Subtract.
The remainder is 8.
+ 3x
– 7 Multiply 5(x – 3). Subtract.
+ 5
(x3 – 4x – 7) ÷ (x – 3) =
10-6 Dividing Polynomials
Recall from Chapter 7 that a polynomial in one variable is written in standard form when the degrees of the terms go from greatest to least.
Remember!
10-6 Dividing Polynomials
Divide (1 – 4x2 + x3) ÷ (x – 2).
Check It Out! Example 5a
(x3 – 4x2 + 1) ÷ (x – 2)
x3 – 4x2 + 0x + 1x – 2)
Write in standard form.Write in long division form.
Use 0x as a placeholder for the x term.
x3 – 4x2 + 0x + 1x – 2)x2 x3 ÷ x = x2
–(–2x2 + 4x)
– 4x –(–4x + 8)
–7
Bring down 0x. – 2x2 ÷ x = –2x.
Multiply –2x(x – 2). Subtract.Bring down 1.Multiply –4(x – 2). Subtract.
–(x3 – 2x2) – 2x2
Multiply x2(x – 2). Subtract.
– 2x
+ 0x
+ 1
– 4
10-6 Dividing Polynomials
Divide (1 – 4x2 + x3) ÷ (x – 2).
Check It Out! Example 5a Continued
(1 – 4x2 + x3) ÷ (x – 2) =
10-6 Dividing Polynomials
Divide (4p – 1 + 2p3) ÷ (p + 1).
Check It Out! Example 5b
(2p3 + 4p – 1) ÷ (p + 1)
2p3 + 0p2 + 4p – 1p + 1)
Write in standard form.
Write in long division form. Use 0p2 as a placeholder for the p2 term.
2p3 + 0p2 + 4p – 1p + 1)2p2
p3 ÷ p = p2
–(–2p2 – 2p)
6p –(6p + 6)
–7
Bring down 4p. –2p2 ÷ p = –2p.
Multiply –2p(p + 1). Subtract.Bring down –1.Multiply 6(p + 1). Subtract.
–(2p3 + 2p2) – 2p2
Multiply 2p2(p + 1). Subtract.
– 2p
+ 4p
– 1
+ 6
10-6 Dividing Polynomials
Lesson Quiz: Part I
Add or Subtract. Simplify your answer.
1.
3.
2.
(12x2 – 4x2 + 20x) ÷ 4x 3x2 – x + 5
x – 2
4. x + 3
2x + 3