Understanding the Polynomial Expansion Problem

The question asks us to expand a given algebraic expression, which is in the form \(4(z + 7)(2z – 1)\), and match it to the standard quadratic form \(Az^2 + Bz + C\). Once we have the expanded form, we need to identify the values of the coefficients \(A\), \(B\), and the constant term \(C\). Finally, we are required to calculate the sum of these values, which is \(A + B + C\).

This involves performing polynomial multiplication and then comparing the resulting expression with the general quadratic equation form to find the coefficients.

Step-by-Step Solution for Expanding the Expression

We start with the given expression: \(4(z + 7)(2z – 1)\).

First, let's expand the product of the two binomials \((z + 7)(2z – 1)\). We can use the FOIL method (First, Outer, Inner, Last):

• First terms: \(z \times 2z = 2z^2\)
• Outer terms: \(z \times -1 = -z\)
• Inner terms: \(7 \times 2z = 14z\)
• Last terms: \(7 \times -1 = -7\)

Now, we combine these terms:

\((z + 7)(2z – 1) = 2z^2 - z + 14z - 7\)

Combine the like terms (\(-z\) and \(14z\)):

\((z + 7)(2z – 1) = 2z^2 + 13z - 7\)

Next, we multiply this result by the factor of 4 that was outside the parentheses:

\(4(2z^2 + 13z - 7)\)

Distribute the 4 to each term inside the parentheses:

• \(4 \times 2z^2 = 8z^2\)
• \(4 \times 13z = 52z\)
• \(4 \times -7 = -28\)

So, the expanded form of the expression is:

\(4(z + 7)(2z – 1) = 8z^2 + 52z - 28\)

Identifying Coefficients A, B, and C

The problem states that \(4(z + 7)(2z – 1) = Az^2 + Bz + C\).

By comparing our expanded form \(8z^2 + 52z - 28\) with the general form \(Az^2 + Bz + C\), we can identify the values of \(A\), \(B\), and \(C\):

• The coefficient of \(z^2\) is \(A\). In our expanded form, the coefficient of \(z^2\) is 8. So, \(A = 8\).
• The coefficient of \(z\) is \(B\). In our expanded form, the coefficient of \(z\) is 52. So, \(B = 52\).
• The constant term is \(C\). In our expanded form, the constant term is -28. So, \(C = -28\).

Calculating the Value of A + B + C

Now that we have the values for \(A\), \(B\), and \(C\), we can calculate their sum:

\(A + B + C = 8 + 52 + (-28)\)

\(A + B + C = 60 - 28\)

\(A + B + C = 32\)

The value of \(A + B + C\) is 32.

Revision Table: Key Steps in Polynomial Expansion

Additional Information on Coefficients and Polynomials

In a polynomial expression like \(Az^2 + Bz + C\), the letters \(A\), \(B\), and \(C\) are called coefficients (except for C, which is also the constant term). They are numerical values that multiply the variables (z² and z) or stand alone (C).

• \(A\) is the coefficient of the \(z^2\) term. If \(A \neq 0\), the polynomial is a quadratic expression.
• \(B\) is the coefficient of the \(z\) term.
• \(C\) is the constant term, which does not have a variable attached.

Understanding coefficients is crucial for working with polynomials, solving equations, and analyzing functions. Equating two polynomial expressions means that the coefficients of corresponding terms must be equal.