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Magazine Chapter 5: Problem 180
Factor each polynomial completely. $$4 q^{2}-28 q+49$$
Short Answer
Expert verified
(2q - 7)^2
Step by step solution
01
- Identify common factors
Examine the polynomial to see if there are any common factors among the terms. In this case, there are no common factors other than 1.
02
- Recognize the trinomial pattern
Observe that the given polynomial is a trinomial in the form of a perfect square. A perfect square trinomial looks like \[a^2 - 2ab + b^2 = (a - b)^2\].
03
- Find the square roots of the first and last terms
The first term, 4q^2, is a perfect square as \[(2q)^2 = 4q^2\]. The last term, 49, is also a perfect square as \[7^2 = 49\].
04
- Check the middle term
The middle term should be \(- 2ab\) where a = 2q and b = 7\.Check if \[- 2 \times 2q \times 7 = -28q\],which matches the middle term of the polynomial.
05
- Write the factored form
Since the polynomial matches the pattern of a perfect square trinomial, it can be factored as follows: \[4 q^{2}-28 q+49 = (2q - 7)^2\].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polynomial Factorization
Polynomial factorization is a method used to express a polynomial as the product of its factors. These factors can be numbers, variables, or other polynomials. Factoring polynomials simplifies expressions and solves equations more easily.
To factor a polynomial, follow these steps:
To factor a polynomial, follow these steps:
- Identify any common factors across all terms 鈥� This means finding the greatest common divisor (GCD) for the coefficients and taking out any shared variables.
- Look for special patterns 鈥� Some polynomials take forms that resemble well-known patterns like difference of squares, sum and difference of cubes, and perfect square trinomials (which we'll discuss in detail below).
- Use polynomial division when necessary 鈥� For more complex polynomials, you might need to use synthetic or polynomial long division to find factors.
Perfect Square Trinomial
A perfect square trinomial is a special type of polynomial. It comes from squaring a binomial. A binomial is simply an expression with two terms, such as (a + b).
When a binomial is squared, the result is a perfect square trinomial. The general form is:
When a binomial is squared, the result is a perfect square trinomial. The general form is:
- (a + b)^2 = a^2 + 2ab + b^2
- (a - b)^2 = a^2 - 2ab + b^2
- Find the square roots of the first and last terms.
- Verify that the middle term matches the form 2ab.
Steps in Factoring
Factoring a polynomial involves several clear steps to ensure that the polynomial is expressed correctly in its simplest form. Here's a detailed breakdown of the steps using our example(4q^2 - 28q + 49):
- **Identify common factors** 鈥� Check if there are any common factors among all terms. In our example, there are no common factors other than 1.
- **Recognize the trinomial pattern** 鈥� Identify if the polynomial fits the perfect square trinomial pattern. For 4q^2 - 28q + 49, it resembles the a^2 - 2ab + b^2 pattern.
- **Find the square roots of the first and last terms** 鈥� For4q^2, the square root is2q and for 49, it is 7.
- **Check the middle term** 鈥� Ensure that the middle term is -2ab (in our case, -2 脳 2q 脳 7, which equals -28q).
- **Write the factored form** 鈥� Since it meets all criteria of a perfect square trinomial, the factorized form is(2q - 7)^2.
