Question

Factor each polynomial. $$ 2 z^{2}+9 z+7 $$

Short answer

The factors are \((2z + 7)(z + 1)\).

Step-by-step solution

- Identify the polynomial coefficients

Identify the coefficients of the polynomial. Here, the polynomial is in the form of \[2z^2 + 9z + 7\]. The coefficients are: a = 2, b = 9, and c = 7.

- Multiply 'a' and 'c'

Multiply the coefficients 'a' and 'c': \[a \times c = 2 \times 7 = 14\].

- Find two numbers that multiply to 14 and add to 'b'

Find two numbers that multiply to 14 (result from Step 2) and add to 9 (coefficient 'b'). The numbers are 2 and 7 since: \[2 \times 7 = 14 \ 2 + 7 = 9\].

- Rewrite the middle term using the two numbers

Rewrite the polynomial by splitting the middle term using the numbers found in Step 3: \[2z^2 + 2z + 7z + 7\].

- Factor by grouping

Group the terms into two pairs and factor out the common factor from each pair: \[2z(z + 1) + 7(z + 1)\].

- Factor out the common binomial factor

Factor out the common binomial factor \((z + 1)\): \[(2z + 7)(z + 1)\].

Key concepts

coefficients

When dealing with polynomials, understanding coefficients is essential. Coefficients are the numerical values in front of the variables in a polynomial. For example, in the polynomial \(2z^2 + 9z + 7\), the terms are \(2z^2\), \(9z\), and \(7\). Here, 2 is the coefficient of \(z^2\), 9 is the coefficient of \(z\), and 7 is a constant term (coefficient of \(z^0\)). Recognizing these helps in performing operations like addition, subtraction, multiplication, and particularly factoring.

The coefficients dictate how the polynomial behaves and are crucial in each step of the factoring process. In our example:

• a = 2
• b = 9
• c = 7

Knowing these, we proceed by finding a product (a * c) and then seek pairs of numbers that multiply to give this product and add to give the middle coefficient (b). This leads us smoothly into the next core concept.

factoring by grouping

Factoring by grouping is a method often used to simplify polynomials. It involves rearranging the terms of the polynomial and factoring out common factors in pairs. This method is particularly useful for polynomials with four terms.

In this exercise, notice how the quadratic polynomial \(2z^2 + 9z + 7\) was rewritten in step 4 by splitting the middle term 9z into two terms, 2z and 7z, resulting in \(2z^2 + 2z + 7z + 7\). The key is to rewrite the polynomial so that each pair of terms has a common factor. The polynomial was then grouped: \[2z^2 + 2z + 7z + 7 \rightarrow (2z^2 + 2z) + (7z + 7)\].

Next, a common factor is factored out from each group: \[(2z(z + 1)) + (7(z + 1))\]. Notice how both groups now contain the common binomial factor \(z + 1\). This technique helps to simplify complex polynomials into a product of binomials, taking us to the final step.

binomial factor

The binomial factor is a key component in the factoring process. A binomial is simply a polynomial with two terms. In our example, after factoring by grouping, the binomial \(z + 1\) appears in both groups. This commonality allows further simplification.

Once you have identified a common binomial, you can factor it out. Here, \[(2z(z + 1)) + (7(z + 1))\] can be factored as \[(z + 1)(2z + 7)\].
This shows that our polynomial \(2z^2 + 9z + 7\) factors into \[(2z + 7)(z + 1)\]. Understanding binomial factors makes it easier to see how polynomials can be broken down into simpler, more manageable parts.

Remember, identifying the common binomial factor is often the last step in factoring a polynomial this way. It consolidates the earlier steps and provides a clear, factored form of the original polynomial.