1 topic 6.3.3 division by polynomials — factoring division by polynomials — factoring

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Topic 6.3.3Topic 6.3.3

Division by Polynomials— Factoring

Division by Polynomials— Factoring

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Lesson

1.1.1

California Standard:10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.

What it means for you:You’ll divide one polynomial by another polynomial by factoring.

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

Key words:• polynomial• monomial• factor• exponent

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Lesson

1.1.1

Now you’re ready to divide a polynomial by another polynomial.

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

The simplest way to do this is by factoring the numerator and the denominator.

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Lesson

1.1.1

Canceling Fractions Helps to Simplify Expressions

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

If a numerical fraction has a common factor in the numerator and denominator, you can cancel it.

For example,

In the same way, if there are common factors in the numerator and denominator of an algebraic fraction, you can cancel them.

This technique’s really useful for dividing polynomials when the polynomials have already been factored.

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Division by Polynomials — FactoringDivision by Polynomials — Factoring

Example 1

Topic

6.3.3

Simplify .

Solution

= 2x + 3

Solution follows…

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Division by Polynomials — FactoringDivision by Polynomials — Factoring

Example 2

Topic

6.3.3

Divide (x + 4)(1 – x)(3x + 2) by (1 – x).

Solution

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

Solution follows…

(x + 4)(1 – x)(3x + 2) ÷ (1 – x)

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Lesson

1.1.1

Guided Practice

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

Solution follows…

Simplify each expression.

1. 2.

3. 4.

5. 6.

= x + 9

= 1

=

= 1

= =

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Lesson

1.1.1

Guided Practice

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

Solution follows…

Simplify each expression.

7.

8.

9. Divide (x + 3)(x + 4) by .

10. Divide by .

= 8x + 32

=

=

=

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Lesson

1.1.1

If It Divides Evenly, the Polynomial Can Be Factored

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

If you can divide a polynomial evenly, that means there is no remainder.

This means that it must be possible to factor the polynomial (and it means that the divisor is a factor of the polynomial).

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Division by Polynomials — FactoringDivision by Polynomials — Factoring

Example 3

Topic

6.3.3

Given that (x + 1) divides evenly into (x2 – 4x – 5), find (x2 – 4x – 5) ÷ (x + 1).

Solution

= (x – 5)

Solution follows…

(x2 – 4x – 5) ÷ (x + 1)

You know that (x + 1) is a factor because you’re told it divides evenlyCancel (x + 1) from the top and bottom

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Lesson

1.1.1

Guided Practice

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

Solution follows…

Simplify the quotients by canceling factors.

11. 12.

13. 14.

15. 16.

= 4 =

= 4z = x + 6

= =

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Lesson

1.1.1

Guided Practice

Division by Polynomials — FactoringDivision by Polynomials — FactoringTopic

6.3.3

Solution follows…

Simplify the quotients by canceling factors.

17. 18.

19.

20. Find the ratio of the surface area to the volume of a cube with side length b.

21. Divide 4x – 12 by x2 – 2x – 3.

=

==

=

= =

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Division by Polynomials — FactoringDivision by Polynomials — Factoring

Independent Practice

Solution follows…

Topic

6.3.3

Name the two factors that would divide into each expression below.

1. x2 + 7x 2. 2x – 8

3. 6a – 15 4. 6x2 + 9x

5. x2 + 8x + 15 6. a2 – 81

x and (x + 7) 2 and (x – 4)

3 and (2a – 5) 3x and (2x + 3)

(x + 3) and (x + 5) (a + 9) and (a – 9)

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Division by Polynomials — FactoringDivision by Polynomials — Factoring

Independent Practice

Solution follows…

Topic

6.3.3

Simplify the quotients by canceling factors.

7. 8. 9.

10. 11. 12.

13. 14.

15.

4a – 8 x2

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Division by Polynomials — FactoringDivision by Polynomials — Factoring

Independent Practice

Solution follows…

Topic

6.3.3

Simplify the quotients by canceling factors.

16. 17.

18.

19. Find the ratio of the surface area to the volume of the rectangular prism shown.

b

2bb

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Topic

6.3.3

Round UpRound Up

Division by Polynomials — FactoringDivision by Polynomials — Factoring

This method’s most useful for “divides evenly” questions.

If a question mentions remainders, the long division method in Topic 6.3.4 is probably better.

There are two methods for polynomial division, and you should use the one that makes the most sense for the question you’re doing.