Question

In Exercises 35–42, fi nd the product. \((7 h+4)^2\)

Short answer

The square of the binomial \( (7h+4)^2 \) is \( 49h^2 + 56h + 16 \)

Step-by-step solution

Identify the variables

First identify the variables in the binomial. It has two parts, 'a' and 'b'. Here, 'a' is '7h' and 'b' is '4'.

Apply the formula

Substitute these values into the formula for the square of a binomial, which is \( (a+b)^2 = a^2 + 2ab + b^2 \). So, \((7h+4)^2 = (7h)^2 + 2*7h*4 + 4^2 \)

Simplify the equation

Simplify the equation to get the final answer. \( (7h)^2 = 49h^2 \), \( 2*7h*4 = 56h \), and \( 4^2 = 16 \). Add these together to get \( 49h^2 + 56h + 16 \)

Key concepts

Polynomial Multiplication

Polynomial multiplication is a fundamental operation in algebra that involves multiplying two or more polynomials to produce a new polynomial. When you multiply polynomials, you follow the distributive property, which entails multiplying each term in the first polynomial by each term in the second polynomial and then combining like terms. An example could be multiplying (3x + 2) by (x - 5), resulting in the polynomial (3x^2 - 13x - 10) after distributing and combining like terms.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that represent mathematical relationships. These expressions can be as simple as a single variable like 'x' or as complex as polynomials with multiple terms, such as (4x^3 - 3x^2 + 2x - 1). To evaluate an algebraic expression, you need to substitute each variable with a given number and perform the operations according to the order of operations.

Squaring Binomials

Squaring binomials is a special case of polynomial multiplication where the same binomial is multiplied by itself. A binomial is an algebraic expression that contains exactly two terms, such as (a + b). The square of a binomial, (a + b)^2, expands to (a^2 + 2ab + b^2), as seen in the textbook exercise. Remember that when squaring a term, you multiply the term by itself, which includes squaring both the numerical coefficient and the variable part.

Math Problem Solving

Math problem solving is the process of understanding a given problem, finding a way to approach it, and working through the steps to arrive at a solution. It often involves a systematic approach: identifying what you know, what you need to find out, deciding on a strategy to solve the problem, implementing the strategy, and then checking to see if the solution makes sense. In algebra, for example, this could mean interpreting the meaning of an equation, graphing it to find solutions, or employing algebraic techniques to solve for a variable. Becoming proficient in math problem solving is not just about knowing the formulas and steps; it's also about developing critical thinking and the ability to logically work through issues to find a solution.