Example Card: Factoring a Perfect-Square Trinomial
Factor perfect-square trinomials in Grade 9 algebra by recognizing the a²+2ab+b²=(a+b)² or a²-2ab+b²=(a-b)² patterns, checking that the first and last terms are perfect squares.
Key Concepts
Let’s see if this trinomial hides a special pattern inside. This relates to our first key idea, factoring perfect square trinomials.
Example Problem Determine if $50x^2 + 40x + 8$ is a perfect square trinomial. If so, factor it.
Step by Step 1. First, we examine the trinomial $50x^2 + 40x + 8$. We can see that the greatest common factor is $2$. $$ 2(25x^2 + 20x + 4) $$ 2. Now, let's see if the expression inside the parentheses, $25x^2 + 20x + 4$, fits the form $a^2 + 2ab + b^2$. The first term is a perfect square: $25x^2 = (5x)^2$. The last term is a perfect square: $4 = 2^2$. 3. We check if the middle term matches $2ab$. Let $a = 5x$ and $b=2$. Then $2ab = 2(5x)(2) = 20x$. This matches the middle term. 4. Since it fits the pattern, we can write it in factored form, remembering the GCF we factored out in step 1. $$ 2(5x + 2)^2 $$ So, the polynomial is a perfect square trinomial.
Common Questions
What is a perfect-square trinomial?
A perfect-square trinomial is a trinomial that results from squaring a binomial. It follows the pattern a² + 2ab + b² = (a + b)² or a² - 2ab + b² = (a - b)². For example, x² + 6x + 9 = (x + 3)².
How do you identify and factor a perfect-square trinomial?
Check three conditions: (1) the first term is a perfect square, (2) the last term is a positive perfect square, and (3) the middle term equals twice the product of the square roots of the first and last terms. If all hold, write as (a ± b)².
How do you factor 4x² - 12x + 9?
First term: 4x² = (2x)². Last term: 9 = 3². Middle term check: 2(2x)(3) = 12x ✓ and the sign is negative. So 4x² - 12x + 9 = (2x - 3)². Verify by expanding: (2x-3)² = 4x² - 12x + 9 ✓