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Chapter 0: Problem 16

Factor the perfect square trinomial.\(9 x^{2}-12 x+4\)

Short Answer

Expert verified
The factor of the perfect square trinomial \(9 x^{2}-12 x+4\) is \((3x - 2)^2\)

Step by step solution

01

Identify the Trinomial

The given trinomial is \(9 x^{2}-12 x+4\)
02

Take the square root of the first and third terms

Take the square root of the first term \(9 x^{2}\) which is \(3x\) and of the third term \(4\), which is \(2\).
03

Formulate the perfect square trinomial

Now the square of binomial can be formed i.e., \((3x - 2)^2\). Verify by expanding the binomial square and compare the result with the original trinomial.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Perfect Square Trinomial
A perfect square trinomial is a special type of trinomial where it can be expressed as the square of a binomial. This means that when you factor the trinomial, you end up with
  • The same value appearing twice as a repeated binomial.
In mathematical terms, a perfect square trinomial has the form \(a^2 + 2ab + b^2\), which factors into \((a + b)^2\).
For example, the trinomial \(9x^2 - 12x + 4\) is a perfect square trinomial. Here, when you factor it, the result is \((3x - 2)^2\). You can see this by taking the square root of the first term \(9x^2\) resulting in \(3x\), and the square root of the last term \(4\) resulting in \(2\).
The middle term \(-12x\) is
  • The product of the two roots \(3x\) and \(-2\), multiplied by 2, which confirms the perfect square trinomial pattern.
Binomial Expansion
The process of expanding a binomial involves multiplying the binomial by itself to get a trinomial. This is known as binomial expansion. When you have a squared binomial like \((a + b)^2\), binomial expansion helps us rewrite it as \(a^2 + 2ab + b^2\).
Here鈥檚 how it works: multiply each term in the binomial by every term in the other binomial:
  • First, multiply \(a\) by \(a\) and \(b\).
  • Then, multiply \(b\) by \(a\) and \(b\).
This will unfold into a trinomial where
  • The first term is \(a^2\),
  • The middle term is \(2ab\),
  • And the last term is \(b^2\).
For instance, if you expand \((3x - 2)^2\), it results in \(9x^2 - 12x + 4\), verifying the perfect square trinomial.
Quadratic Expressions
Quadratic expressions are algebraic expressions of degree 2, typically in the form \(ax^2 + bx + c\). A perfect square trinomial is a type of quadratic expression.
This specific structure suggests that the expression can be squared to achieve or revert to a simpler binomial.
Understanding quadratic expressions involves recognizing:
  • The coefficient of \(x^2\) (the leading term that determines the form of expansion),
  • The linear coefficient (the term affecting symmetry of the graph if plotted),
  • The constant term (that shifts the entire graph up or down).
In our example \(9x^2 - 12x + 4\),
  • 9 is the leading coefficient indicating perfect squares are part of the expression,
  • -12x reflects the type and direction of linear shifts,
  • while 4 represents vertical translation.
Recognizing these properties is essential in factoring and solving equations related to quadratics. This lays the foundation for efficiently working with deeper algebra concepts.

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Most popular questions from this chapter

Simplify the expression.\(\left(16 x^{8} y^{4}\right)^{3 / 4}\)

Factor the sum or difference of cubes.\(y^{3}+125\)

Factor out the common factor.\(3 x+6\)

Factor the difference of two squares.\(x^{2}-49 y^{2}\)

Simplify the expression.\(\sqrt[5]{64 y^{-5}}\)
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