4 2. Look for special patterns = (x + 4)(x– 4)x2 – 16x2 + 14x + 49= (x + 7)2There are other special patterns that are also worth remembering
5 SPECIAL FACTORING PATTERNS NAME Difference ofTwo cubesSum of two cubesx3 + y3 =(x + y)(x2 –xy+ y2)PATTERNx3 - y3 =(x - y)(x2 +xy+ y2)x3 + 8= (x + 2)(x2 -2x+ 4)EXAMPLE8x3 – 1(2x - 1)(4x2 +2x+ 1)
6 3.Factoring by grouping 1. Begin by factoring out the GCF. 1. 5x3+2x2-40x-16None2. Arrange the four terms so that the first two terms and the last two terms have common factors.2. (5x3+2x2)+(-40x-16)3. If the coefficient of the third term is negative, factor out a negative coefficient from the last two terms.3. (5x3+2x2)-(40x+16)4. Use the reverse of the distributive property to factor each group of two terms.4. x2(5x+2)-8(5x+2)5. Now factor the GCF from the result of step 4 as done in the previous section.5. (5x+2)(x2-8)
7 Factoring using quadratics x4 + 3x2 -4The following steps can be used to solve equations that are quadratic in form:1. Let u equal a function of the original variable (normally the middle term)2. Substitute u into the original equation so that it is in the form au2 + bu + c3. Factor the quadratic equation using the methods learned earlier4. Replace u with the expression of the original variable.5. Factor again if necessary.1. u=x22. u2+3u-43. (u+4)(u-1)4. (x2+4)(x2-1)5. (x2+4)(x-1)(x+1)
8 Solving Polynomials Remember Finding zeros, solutions, and roots are different ways of saying the same thing.So…After you factor the polynomial, set it equal to 0.Then solve the polynomial.
9 Find the real-number solutions x4 + 3x2 -4=0(x2+4)(x-1)(x+1)=0(x2+4=0, not real(x-1)=0 x=1(x+1)=0 x=-1