Problem 63 - Algebra and Trigonometry with Analytic Geometry

Question

Factor the polynomial. $$ 25 z^{2}+30 z+9 $$

Step-by-Step Solution

Answer

The polynomial factors as \((5z + 3)^2\).

Step 1: Identify the Quadratic Expression

The given polynomial is \(25z^2 + 30z + 9\). This is a quadratic expression in the form \(az^2 + bz + c\) where \(a = 25\), \(b = 30\), and \(c = 9\).

Step 2: Check for Perfect Square Trinomial

To verify if the expression is a perfect square trinomial, check if both the first term \(25z^2\) and the last term \(9\) are perfect squares. We find \(\sqrt{25z^2} = 5z\) and \(\sqrt{9} = 3\). Check if twice the product of these roots equals the middle term: \(2 \times 5z \times 3 = 30z\), which matches the middle term.

Step 3: Write the Expression as a Square of a Binomial

Since \(25z^2 + 30z + 9\) is a perfect square trinomial, it can be expressed as the square of a binomial. The expression can be rewritten as \((5z + 3)^2\) because \((5z)^2 = 25z^2\), \(2 \times 5z \times 3 = 30z\), and \(3^2 = 9\).

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial of degree 2, typically written in the form \( ax^2 + bx + c \), where \( a, b, \) and \( c \) are constants, and \( a eq 0 \). In simpler terms, it's an expression with a squared term as its highest power of the variable. Quadratic expressions can show up in various places in math, from physics problems to geometry and economics.

• The coefficient \( a \) is the number in front of the squared term, which in our exercise is 25, making it \( 25z^2 \).
• The coefficient \( b \) is the number in front of the linear term, here 30, making it \( 30z \).
• The constant term \( c \) is the standalone number, which in this case is 9.

Recognizing quadratic expressions is important because they have unique properties and solutions, such as factoring, which help solve many types of problems.

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression that can be expressed as the square of a binomial. This means if an expression fits this formula, \( (px + q)^2 = p^2x^2 + 2pqx + q^2 \), then it is a perfect square trinomial.

• The first term \( p^2x^2 \) is formed by squaring the variable term.
• The last term \( q^2 \) is the square of the constant term.
• The middle term \( 2pqx \) equals twice the product of the terms in the binomial.

For the expression \( 25z^2 + 30z + 9 \), both \( 25z^2 \) and 9 are perfect squares (since \((5z)^2 = 25z^2\) and \(3^2 = 9\)). The middle term, 30z, equals two times the product of these roots, \( 2 \times 5z \times 3 \), confirming it's a perfect square trinomial. Recognizing when a quadratic is a perfect square trinomial allows us to rewrite it simply, making it easier to work with.

Binomial Square

A binomial square is what you get when you take a binomial and square it. This is represented as \((px + q)^2\), which expands to \(p^2x^2 + 2pqx + q^2\).
In our task, we transform the quadratic expression \(25z^2 + 30z + 9\) into a binomial square.

• We identify \(5z\) as \(p\) and \(3\) as \(q\).
• Thus, \((5z + 3)^2\) becomes the binomial square that matches the original trinomial.
• This transformation simplifies the factorization process, making solving equations or understanding graph relationships much easier.

Using the concept of a binomial square helps in numerous mathematical problems, particularly in simplifying expressions and solving quadratic equations by factoring. By recognizing \((5z + 3)^2\), it tells us immediately how the expression behaves and what its roots might be.