Find a common monomial factor

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EXAMPLE 1
Find a common monomial factor
Factor the polynomial completely.
a. x3 2x2 15x
Factor common monomial.
x(x2 2x 15)
x(x 5)(x 3)
Factor trinomial.
2y3(y2 9)
b. 2y5 18y3
Factor common monomial.
2y3(y 3)(y 3)
Difference of two squares
4z2(z2 4z 4)
c. 4z4 16z3 16z2
Factor common monomial.
4z2(z 2)2
Perfect square trinomial
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EXAMPLE 2
Factor the sum or difference of two cubes
Factor the polynomial completely.
a. x3 64
x3 43
Sum of two cubes
(x 4)(x2 4x 16)
2z2(8z3 125)
b. 16z5 250z2
Factor common monomial.
Difference of two cubes
2z2(2z 5)(4z2 10z 25)
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EXAMPLE 3
Factor by grouping
Factor the polynomial x3 3x2 16x 48
completely.
x2(x 3) 16(x 3)
x3 3x2 16x 48
Factor by grouping.
(x2 16)(x 3)
Distributive property
(x 4)(x 4)(x 3)
Difference of two squares
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EXAMPLE 4
Factor polynomials in quadratic form
Factor completely (a) 16x4 81 and (b) 2p8
10p5 12p2.
a. 16x4 81
Write as difference of two squares.
(4x2)2 92
(4x2 9)(4x2 9)
Difference of two squares
(4x2 9)(2x 3)(2x 3)
Difference of two squares
Factor common monomial.
b. 2p8 10p5 12p2
2p2(p6 5p3 6)
Factor trinomial in quadratic form.
2p2(p3 3)(p3 2)
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EXAMPLE 5
Standardized Test Practice
SOLUTION
3x5 15x 18x3
Write original equation.
3x5 18x3 15x 0
Write in standard form.
3x(x4 6x2 5) 0
Factor common monomial.
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EXAMPLE 5
Standardized Test Practice
3x(x2 1)(x2 5) 0
Factor trinomial.
3x(x 1)(x 1)(x2 5) 0
Difference of two squares
Zero product property
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EXAMPLE 1
Use polynomial long division
Divide f (x) 3x4 5x3 4x 6 by x2 3x 5.
SOLUTION
Write polynomial division in the same format you
use when dividing numbers. Include a 0 as the
coefficient of x2 in the dividend. At each stage,
divide the term with the highest power in what is
left of the dividend by the first term of the
divisor. This gives the next term of the quotient.
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EXAMPLE 1
Use polynomial long division
3x2 4x 3
Multiply divisor by 3x4/x2 3x2
Subtract. Bring down next term.
4x3 15x2 4x
Multiply divisor by 4x3/x2 4x
Subtract. Bring down next term.
3x2 16x 6
Multiply divisor by 3x2/x2 3
25x 9
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EXAMPLE 1
Use polynomial long division
CHECK
You can check the result of a division problem by
multiplying the quotient by the divisor and
adding the remainder. The result should be the
dividend.
(3x2 4x 3)(x2 3x 5) ( 25x 9)
3x2(x2 3x 5) 4x(x2 3x 5) 3(x2 3x
5) 25x 9
3x4 9x3 15x2 4x3 12x2 20x 3x2 9x
15 25x 9
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EXAMPLE 2
Use polynomial long division with a linear divisor
Divide f (x) x3 5x2 7x 2 by x 2.
x2 7x 7
Multiply divisor by x3/x x2.
7x2 7x
Subtract.
Multiply divisor by 7x2/x 7x.
7x 2
Subtract.
Multiply divisor by 7x/x 7.
16
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EXAMPLE 3
Use synthetic division
Divide f (x) 2x3 x2 8x 5 by x 3 using
synthetic division.
SOLUTION
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EXAMPLE 4
Factor a polynomial
Factor f (x) 3x3 4x2 28x 16 completely
given that x 2 is a factor.
SOLUTION
Because x 2 is a factor of f (x), you know that
f ( 2) 0. Use synthetic
division to find the other factors.
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EXAMPLE 4
Factor a polynomial
Use the result to write f (x) as a product of two
factors and then factor completely.
f (x) 3x3 4x2 28x 16
Write original polynomial.
(x 2)(3x2 10x 8)
Write as a product of two factors.
(x 2)(3x 2)(x 4)
Factor trinomial.
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EXAMPLE 5
Standardized Test Practice
SOLUTION
Because f (3) 0, x 3 is a factor of f (x).
Use synthetic division.
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EXAMPLE 5
Standardized Test Practice
Use the result to write f (x) as a product of two
factors. Then factor completely.
f (x) x3 2x2 23x 60
(x 3)(x2 x 20)
(x 3)(x 5)(x 4)
The zeros are 3, 5, and 4.