7. Factoring Special Products
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How can we use the known special products patterns?
See solution.
Since we know the special product patterns we can identify if a polynomial fits in one of these, to be later factored accordingly.
The process of squaring a binomial gives as result a perfect square trinomial. (a+b)^2 =a^2+2ab+ b^2 (a-b)^2 =a^2-2ab+ b^2 Therefore any perfect square trinomial can be factored as the square of a binomial. Note that a trinomial has to satisfy two conditions to be a perfect square trinomial.
Let's consider the example trinomial shown below. x^2+8x+16 Note that x^2 and 16=4^2 are perfect squares whose square roots are x and 4. Hence, the first condition is satisfied. Furthermore, the middle term 8x equals two times the product of the square roots of the others. 8x &= 2* x*4 8x &= 2* 4* x 8x &= 8x ✓ Therefore the second condition is also satisfied, and we can factor the trinomial as the square of a binomial. x^2+8x+16 ⇔ (x+4)^2 Here, the binomial is a sum of two terms. If the middle term of the trinomial was negative we would have used a negative sign for the square of the binomial instead. x^2 - 8x+16 ⇔ (x - 4)^2