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Holt Algebra 1
12-6 Dividing Polynomials12-6 Dividing Polynomials
Holt Algebra 1
Warm UpWarm Up
Lesson PresentationLesson Presentation
Lesson QuizLesson Quiz
Holt Algebra 1
12-6 Dividing Polynomials
Warm Up
Divide.
1. m2n ÷ mn4 2. 2x3y2 ÷ 6xy
3. (3a + 6a2) ÷ 3a2b
Factor each expression.
4. 5x2 + 16x + 12
5. 16p2 – 72p + 81
Holt Algebra 1
12-6 Dividing Polynomials
Divide a polynomial by a monomial or binomial.
Objective
Holt Algebra 1
12-6 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.
Holt Algebra 1
12-6 Dividing Polynomials
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.
Holt Algebra 1
12-6 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.
Holt Algebra 1
12-6 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.
Simplify.
Holt Algebra 1
12-6 Dividing Polynomials
Division of a polynomial by a binomial is similar to division of whole numbers.
Holt Algebra 1
12-6 Dividing Polynomials
Example 2A: Dividing a Polynomial by a Binomial
Divide.
x + 5
Factor the numerator.
Divide out common factors.
Simplify.
Holt Algebra 1
12-6 Dividing Polynomials
Example 2B: Dividing a Polynomial by a Binomial
Divide.
Factor both the numerator and denominator.
Divide out common factors.
Simplify.
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 Dividing Polynomials
Check It Out! Example 2a
Divide.
k + 5
Factor the numerator.
Divide out common factors.
Simplify.
Holt Algebra 1
12-6 Dividing Polynomials
Check It Out! Example 2b
Divide.
b – 7
Factor the numerator.
Divide out common factors.
Simplify.
Holt Algebra 1
12-6 Dividing Polynomials
Check It Out! Example 2c
Divide.
s + 6
Factor the numerator.
Divide out common factors.
Simplify.
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 Dividing Polynomials
Holt Algebra 1
12-6 Dividing Polynomials
Example 3A: Polynomial Long Division
Divide using long division.
(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
Holt Algebra 1
12-6 Dividing Polynomials
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.
Holt Algebra 1
12-6 Dividing Polynomials
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
Holt Algebra 1
12-6 Dividing Polynomials
Example 3A Continued
Check: Multiply the answer and the divisor.
(x + 3)(x + 7)
x2 + 3x + 7x + 21
x2 + 10x + 21
Holt Algebra 1
12-6 Dividing Polynomials
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
Holt Algebra 1
12-6 Dividing Polynomials
Example 3B Continued
Check: Multiply the answer and the divisor.
(x + 2)(x – 4)
x2 + 2x – 4x – 8
x2 – 2x + 8
Holt Algebra 1
12-6 Dividing Polynomials
Check 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
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 Dividing Polynomials
Check 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
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 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.
Holt Algebra 1
12-6 Dividing Polynomials
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
Holt Algebra 1
12-6 Dividing Polynomials
Example 4 Continued
Divide (3x2 + 19x + 26) ÷ (x + 5)
Write the quotient with the remainder.
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 Dividing Polynomials
Check It Out! Example 4a Continued
Divide.
Write the remainder as a rational expression using the divisor as the denominator.
Holt Algebra 1
12-6 Dividing Polynomials
Check 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 +
Holt Algebra 1
12-6 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.
Holt Algebra 1
12-6 Dividing Polynomials
Example 5: Dividing Polynomials That Have a Zero Coefficient
Divide (x3 – 7 – 4x) ÷ (x – 3).
(x3 – 4x – 7) ÷ (x – 3) Write in standard format.
x3 + 0x2 – 4x – 7 )x – 3Write in long division form.
Use 0x2 as a placeholder for the x2 term.
Holt Algebra 1
12-6 Dividing Polynomials
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.
Holt Algebra 1
12-6 Dividing Polynomials
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) =
Holt Algebra 1
12-6 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!
Holt Algebra 1
12-6 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 format.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
Holt Algebra 1
12-6 Dividing Polynomials
Divide (1 – 4x2 + x3) ÷ (x – 2).
Check It Out! Example 5a Continued
(1 – 4x2 + x3) ÷ (x – 2) =
Holt Algebra 1
12-6 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 format.
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
Holt Algebra 1
12-6 Dividing Polynomials
Check It Out! Example 5b Continued
(2p3 + 4p – 1) ÷ (p + 1) =
Holt Algebra 1
12-6 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
Holt Algebra 1
12-6 Dividing Polynomials
Lesson Quiz: Part II
Divide using long division.
5.
6. (8x2 + 2x3 + 7) (x + 3)
(x2 + 4x + 7) (x + 1)
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