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Factoring the GCF

Factoring the GCF. Algebra 1: Section 10.8. What’s the GCF?. The Greatest Common Factor (GCF) of a polynomial is the term ax n such that a is the greatest common factor of all of the coefficients and x n is smallest power of x in the polynomial.

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Factoring the GCF

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  1. Factoring the GCF Algebra 1: Section 10.8

  2. What’s the GCF? • The Greatest Common Factor (GCF) of a polynomial is the term axn such that a is the greatest common factor of all of the coefficients and xn is smallest power of x in the polynomial. • The GCF of 4x5 – 2x4 + 6x2 is 2x2 since 2 is the largest number that is a factor of 4, -2, and 6 and x2 is the smallest power of x.

  3. Factoring Out the GCF • To factor out the GCF of a polynomial; • Identify the GCF, • Factor it out, • Rewrite the reduced polynomial inside parenthesis. • It may be possible to factor the reduced polynomial even further using the technique we have already learned.

  4. Factoring Out the GCF • Factor 14x4 – 21x2 completely. • The GCF is 7x2. • Reduce: 7x2(2x2 – 3). • Factor 2x2 – 8 completely. • The GCF is 2. • Reduce: 2(x2 – 4). • Factor: 2(x + 2)(x – 2).

  5. Factoring Out the GCF • Factor 2x2 + 8 completely. • The GCF is 2. • Reduce: 2(x2 + 4). • Factor: 2(x2 + 4) cannot be factored further. • Factor 4x3 + 20x2 + 24x completely. • The GCF is 4x. • Reduce: 4x(x2 + 5x + 6). • Factor: 4x(x + 3)(x + 2).

  6. Factoring Out the GCF • Solve 8x3 – 18x = 0. • The GCF is 2x. • Reduce: 2x(4x2 – 9) = 0. • Factor: 2x(2x + 3)(2x – 3) = 0. • Solve: x = 0, -3/2, 3/2.

  7. Factoring Out the GCF • Solve -30x4 + 58x3 – 24x2 = 0. • The GCF is -2x2. • Reduce: -2x2(15x2 – 29x + 12) = 0. • Factor: -2x2(15x2 – 29x + 12) = 0 • Split the middle: -2x2(15x2 – 9x – 20x + 12) = 0 • Factor: -2x2(3x(5x – 3) – 4(5x – 3)) = 0 • Factor: -2x2(3x – 4)(5x – 3) = 0. • Solve: x = 0, 4/3, 3/5.

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