Answer:
4x² - 12x + 9
4a² - 10a + 25
Explanation:
If you want to have some fun with math, you should try to find perfect square trinomials. These are polynomials that look like this:
a² + 2ab + b² = (a + b)²
or
a² - 2ab + b² = (a - b)²
They are called perfect square trinomials because they are the squares of binomials. For example, x² + 2x + 1 is a perfect square trinomial because it is the same as (x + 1)².
How do you spot a perfect square trinomial? Here are some tips:
- The first and last terms must be perfect squares. For example, 4x² and 9 are perfect squares because they are 2x times 2x and 3 times 3.
- The middle term must be double the product of the square roots of the first and last terms. For example, 6x is double of 2x times 3, which are the square roots of 4x² and 9.
Let's practice with some examples:
4x²- 12x + 9
This is a perfect square trinomial because:
- The first and last terms are perfect squares: 4x² = (2x)² and 9 = (3)²
- The middle term is double the product of the square roots of the first and last terms: -12x = 2 * -2x * 3
We can write this polynomial as (2x - 3)².
16x² + 24x - 9
This is not a perfect square trinomial because:
- The first and last terms are perfect squares: 16x² = (4x)² and 9 = (3)²
- The middle term is not double the product of the square roots of the first and last terms: 24x ≠ 2 * -4x * -3
We cannot write this polynomial as a square of a binomial.
4a² - 10a + 25
This is a perfect square trinomial because:
- The first and last terms are perfect squares: 4a² = (2a)² and 25 = (5)²
- The middle term is double the product of the square roots of the first and last terms: -10a = 2 * -2a * -5
We can write this polynomial as (2a - 5)².
36b² - 24b - 16
This is not a perfect square trinomial because:
- The first and last terms are perfect squares: 36b² = (6b)² and 16 = (4)²
- The middle term is not double the product of the square roots of the first and last terms: -24b ≠ 2 * -6b * -4
We cannot write this polynomial as a square of a binomial.
Now you know how to find perfect square trinomials. They are fun, right?
The polynomials that are perfect square trinomials are:
a. 4x² - 12x + 9 = (2x - 3)^2 c. 4a² - 10a + 25 = (2a - 5)²