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Wize High School Grade 12 Pre-Calculus Textbook > Polynomial Functions

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
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Factor 8x3−125y38x^3-125y^3