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Comprehensive Guide to Polynomial Factoring Techniques

This guide reviews essential methods for factoring polynomials, focusing on techniques such as factoring out common terms, binomial factorization, and applying the difference and sum of squares and cubes. It covers simplifying expressions like (x^2 + 2y)(x^3 - 3x^2 + 5) and -6a(4a^2 + 7b^2 - 3ab). Learn to use grouping to factor polynomials effectively and the application of second-degree polynomial formulas for accurate binomial factorization. Includes examples and practice problems for a thorough understanding.

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Comprehensive Guide to Polynomial Factoring Techniques

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  1. 1. Simplify (x2 + 2y)(x3-3x2+5) 2. Simplify -6a(4a2 + 7b2 – 3ab) 3. What is the degree of the answer to number 2. Warm up

  2. Objective: To review all previously learned methods of factoring Factoring

  3. Factoring a common term (monomial) • When a polynomial has a factor (either a constant or a variable) that is common to all terms it should be the first thing factored out of the polynomial. • (x3 + x) factors to x(x2 + 1) Factoring

  4. 4x2 –x 3x4 – 9x2 3m(2x-3y)-n(2x-3y) Factoring

  5. Factoring by Grouping – sometimes once you group the terms of the polynomial you will find that you can then factor out common terms. • 3xy+2y + 3xz +2z try grouping the terms containing y together and the terms containing z together • (3xy + 2y) + (3xz + 2z) factor out common terms y(3x +2) + z(3x + 2) factor out common binomials (3x+2)(y +z) Factoring

  6. Try: • 2m3n + m2 + 2mn2 +n • 2a2– 4ab2– ab + 2b3 Factoring

  7. Factoring Second Degree Polynomials • Polynomials in the form x2 + bx + c factor to 2 binomials (x + __)(x + __) • The last numbers in the binomial are found by finding the factors of c that add up to b. • Ex: x2 + 7x +12 1 12 = 13 2 6 = 8 3 4 = 7 Factoring

  8. x2 – 7x + 10 x2 – 3x - 4 Factoring

  9. Factoring Second Degree Polynomials • Polynomials in the form ax2 + bx + c factor to 2 binomials (_x + __)(_x + __) • You must have factors of a before x and factors of c at the end of each binomial. • Ex: 3x2 + 9x + 6 (3x + __)(x + __) look for factors of 6 that will make the outer and inner multiply to 9. (3x + 3)(x + 2) factoring

  10. Another approach to that type of polynomial: • 3x2 + 11x + 6 Put the 3x at the beginning of (3x +__)(3x +__) each binomial Now multiply the 6 by 3 - find factors of 18 that add up to 11 (3x + 9)(3x + 2) Now the first binomial can be factored again, making it (x + 3) so that is what the final factor should look like: (x + 3)(3x + 2) Factoring

  11. Try: • 3x2 – 16x + 21 • 2x2 + 3x – 9 Factoring

  12. Difference of Two Squares a2 – b2 • Factors to (a + b)(a-b) • Ex: x2 – 25 (x + 5)(x – 5) Factoring

  13. 9s2 – 49t2 x4 - 1 Factoring

  14. Sum & Difference of Two Cubes • a3 + b3 = (a + b)(a2 – ab + b2) • a3 – b3 = (a – b)(a2 + ab + b2) • Ex: x3 + 8 (x + 2)(x2 – 2x + 4) x3 – 27 (x – 3)(x2 + 3x + 9) Factoring

  15. x3 – 1 64m3 + 125 Factoring

  16. All these methods could be combined: • 2x3 – 8x factor out a common monomial • 2x(x2 – 4) difference of 2 squares • 2x(x-2)(x+2) once you are sure that each factor is prime – you are done Factoring

  17. x3 + 5x2 – 6x 2x3 – 2x2y – 4xy2 -3x(x+1) + (x + 1)( 2x2 + 1) Factoring

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