Problem 15 - Algebra A Combined Function

Question

Factor each trinomial completely. $$ 16 a^{2}-24 a+9 $$

Step-by-Step Solution

Answer

The trinomial factors as \((4a - 3)^2\).

Step 1: Recognize the trinomial as a perfect square

Observe the trinomial \( 16a^2 - 24a + 9 \). Notice that it resembles the form \( (A - B)^2 = A^2 - 2AB + B^2 \).

Step 2: Identify A and B

The expression can be matched to \( (4a)^2 - 2(4a)(3) + (3)^2 \). Here, \( A = 4a \) and \( B = 3 \).

Step 3: Express the trinomial as a squared binomial

Rewrite the expression: \( (4a - 3)^2 \). The trinomial \( 16a^2 - 24a + 9 \) is a perfect square, factored as \( (4a - 3)^2 \).

Step 4: Verify the factorization

Expand \( (4a - 3)^2 \) to check: \( (4a - 3)(4a - 3) = 16a^2 - 12a - 12a + 9 = 16a^2 - 24a + 9 \). The factorization is verified.

Key Concepts

Perfect Square Trinomials

Understanding perfect square trinomials is the first step in mastering polynomial factorization. A perfect square trinomial is an algebraic expression of the form \( A^2 - 2AB + B^2 \) or \( A^2 + 2AB + B^2 \). These trinomials are derived from the square of a binomial. Recognizing this pattern helps in simplifying complex expressions.

In our exercise, the trinomial \( 16a^2 - 24a + 9 \) fits perfectly into the pattern \( (A - B)^2 = A^2 - 2AB + B^2 \). Each term matches with a component in the formula:

• \( A^2 = 16a^2 \) (meaning \( A = 4a \))
• \( -2AB = -24a \)
• \( B^2 = 9 \) (meaning \( B = 3 \))

Recognizing these relationships makes it easier to see how the trinomial can be rewritten as a squared binomial.

Squared Binomials

Squared binomials are fundamental in algebra and play a critical role in transforming trinomials. A squared binomial is simply the square of a two-term expression, like \((A - B)^2\) or \((A + B)^2\). To make a trinomial into a squared binomial, you identify the terms in the trinomial that match \( A^2 \), \( 2AB \), and \( B^2 \).

In the initial example \( 16a^2 - 24a + 9 \), it can be rewritten as \((4a - 3)^2\). Start by identifying \( A = 4a \) and \( B = 3 \), then apply the squared binomial formula:

• First, ensure each term in \( A^2 - 2AB + B^2 \) matches \( (A - B)^2 \). For our example, the squared form gives \((4a)^2 - 2\times4a\times3 + 3^2\).
• Finally, check that the trinomial reverts back to its original form when expanded: \((4a - 3)(4a - 3) = 16a^2 - 24a + 9\).

Understanding squared binomials is essential for simplifying polynomial expressions through factorization.

Polynomial Factorization

Polynomial factorization involves breaking down expressions into simpler "factor" terms that, when multiplied together, give the original polynomial. This process makes solving equations and simplifying expressions more efficient. One common form of polynomial factorization is transforming a trinomial into a squared binomial.

By recognizing standard patterns (like perfect square trinomials), you can quickly identify factors. In \( 16a^2 - 24a + 9 \), the trinomial is factored into \((4a - 3)^2\). Here's how to approach it:

• First, observe if the trinomial fits into a recognizable pattern: perfect square trinomial or difference of squares.
• Then, express the trinomial in terms of a squared binomial if it acknowledges the perfect square trinomial pattern.
• Lastly, verify your factorization by expanding the squared binomial and checking it returns to the original expression.

This technique not only simplifies the polynomial but also helps to confirm solutions to polynomial equations.