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Wize University Calculus 1 Textbook > Pre-Calculus (Review)

Factoring

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Factoring Review

Factoring a polynomial function, f(x)f(x), can help us determine where the xx-intercepts of of the function are.

Factoring Trinomials in the form y=ax2+bx+c\blue{y=ax^2+bx+c} Where a=1\blue {a=1}

Example 1
Factor f(x)=x2+5x+6f(x)=x^2+5x+6
  1. Find two numbers rr and ss that have a product of a×ca\times c (the first and last coefficients) and sum of bb (the middle coefficient).
Product of 1×6=61\times 6=6 and Sum of 55
The numbers are +2\bold{\orange{+2}} and +3\bold{\orange{+3}}
  1. Rewrite f(x)f(x) as a product of two binomials f(x)=(x+r)(x+s)f(x)=(x+r)(x+s)
Therefore, f(x)=(x+2)(x+3)f(x)=(x\bold{\orange{+2}})(x\bold{\orange{+3}})
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Factoring Trinomials in the form y=ax2+bx+c\blue{y=ax^2+bx+c} Where a≠1\blue {a\neq1}

Example 2
Factor f(x)=2x2−x−1f(x)=2x^2-x-1
  1. Find two numbers rr and ss that have a product of a×ca\times c (the first and last coefficients) and sum of bb (the middle coefficient).
Product of 2×(−1)=−22\times (-1)=-2 and Sum of −1-1
The numbers are −2\bold{\orange{-2}} and +1\bold{\orange{+1}}
  1. Rewrite up the middle term bxbx as+rx+sx+rx+sx
f(x)=2x2−2x+1x−1f(x)=2x^2\bold{\orange{-2}}x\bold{\orange{+1}}x-1
  1. Group the terms together in pairs then factor out the common factors
f(x)=(2x2−2x)+(1x−1)f(x)=(2x^2-2x)+(1x-1) f(x)=2x(x−1)+1(x−1)f(x)=2x\bold{\purple{(x-1)}}+1\bold{\purple{(x-1)}} f(x)=(2x+1)(x−1)f(x)=(2x+1)(x-1)
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Factoring a Difference of Squares y=a2x2−b2:\blue{y=a^2x^2-b^2:}

If f(x)=a2x2−b2, f(x)=a^2x^2-b^2,~ then f(x)=(ax−b)(ax+b) f(x)=(ax-b)(ax+b)~ is in factored form.

Example 3
Factor f(x)=4x2−49f(x)=4x^2-49
f(x)=4x2−49=(2x−7)(2x+7)\begin{array}{r l} f(x)=&4x^2-49\\ =&(2x-7)(2x+7) \end{array}

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Factoring a Difference & Sum of Cubes

Factoring a Difference of Cubes y=a3−b3\blue{y=a^3-b^3}

If f(x)=a3−b3f(x)=a^3-b^3, then f(x)=(a−b)(a2+ab+b2)f(x)=(a-b)(a^2+ab+b^2) is in factored form

Example 1
Factor f(x)=x3−8f(x)=x^3-8
f(x)=x3−8=(x−2)(x2+2x+4)\begin{array}{r l} f(x)=&x^3-8\\ =&(x-2)(x^2+2x+4) \end{array}


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Factoring a Sum of Cubes y=a3+b3\blue{y=a^3+b^3}

If f(x)=a3+b3, f(x)=a^3+b^3,~then f(x)=(a+b)(a2−ab+b2)f(x)=(a+b)(a^2-ab+b^2) is in factored form

Example 2
Factor f(x)=8x3+27f(x)=8x^3+27
f(x)=8x3+27=(2x+3)(4x2+6x+9)\begin{array}{r l} f(x)=&8x^3+27\\ =&(2x+3)(4x^2+6x+9) \end{array}


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Example: Factoring

Factor 7x2−31x−207x^2-31x-20

7x2−31x−20=7x2−35x+4x−20=(7x2−35x)+(4x−20)=7x(x−5)+4(x−5)=(x−5)(7x+4)\begin{array}{r c l} 7x^2-31x-20&=&7x^2-35x+4x-20\\\\ &=&(7x^2-35x)+(4x-20)\\\\ &=&7x(x-5)+4(x-5)\\\\ &=&(x-5)(7x+4) \end{array}
Factor x2−7x−18x^2-7x-18
Factor 2x2+17x+212x^2+17x+21
checklist
Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
Factor 8x3−125y38x^3-125y^3
Extra Practice
Factoring (Review)
Practice: Factoring
Factor 2x2+17x+212x^2+17x+21
Factoring (Review)
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Factor 625x4−16y4625x^4-16y^4
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Factor 27x3−y327x^3-y^3
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Practice: Factoring
Factor 5x2−31x+65x^2-31x+6
Factoring (Review)
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Factor 3x2−13x−103x^2-13x-10
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Practice: Factoring
Factor x2−10x+21x^2-10x+21
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Practice: Factoring
Factor x2−9x−10x^2-9x-10
Factoring (Review)
Factor 8x3−125y38x^3-125y^3
Factoring (Review)
Factor 2x2+17x+212x^2+17x+21
Factoring
Factor x2−7x−18x^2-7x-18
Factoring
Practice: Factoring Harder Trinomials
Factor the following polynomials fully.
a) 36x2+39x−1236x^2+39x-12
Factoring
Practice: Factoring
Factor the quadratic expression 3x2−7x+23x^2-7x+2.