Question

Find the product. $$(5-w)(12+3 w)$$

Short answer

The product of the polynomials (5-w) and (12+3w) is -3w^2 + 3w + 60

Step-by-step solution

Distribute the first term of the first parenthesis

Multiply the first term of the first parenthesis (which is 5) with each term inside the second one. This results in \(5 * 12 = 60\) and \(5 * 3w = 15w\).

Distribute the second term of the first parenthesis

Then, multiply the second term of the first parenthesis (which is -w) with each term in the second one. This results in \(-w * 12 = -12w\) and \(-w * 3w = -3w^2\).

Combine like terms

Next, combine the like terms together. The common terms in the expressions are 15w and -12w which together result in \(15w -12w = 3w\).

Write down the Polynomial Result

Now, write down the result by combining all the results of multiplication you found in step 1 and 2. The final expression should be arranged starting with the term of highest degree. The answer is \(-3w^2 + 3w + 60\).

Key concepts

Understanding the Distributive Property

When it comes to polynomial multiplication, the distributive property is a crucial tool that allows us to multiply a single term by each term within a polynomial. Think of it as an efficient way to ensure every part of the expression is coupled with the term.

The distributive property follows the rule: \( a(b + c) = ab + ac \). By applying this to the given exercise, we distribute the terms of the first polynomial \(5-w\) onto each term of the second polynomial \(12+3w\). This means we'll perform two distributions: First, multiply 5 by each term inside the second parenthesis, and then multiply -w by each term.

This property simplifies complex polynomial multiplication to more manageable single-term multiplications, paving the way to a solvable expression. It's akin to sharing a batch of cookies evenly among friends - everyone gets an equal portion.

Combining Like Terms

Once the distributive property is applied, the next important process in polynomial multiplication is combining like terms. This means adding or subtracting terms that have the same variable and the same exponent. It's like finding pairs of matching socks in a drawer and grouping them together.

In the context of our exercise, after distributing, we end up with a mix of terms including \(3w\) and \(15w\), both of which include the variable 'w' raised to the same power (in this case, 1). These are like terms, and we can combine them by simply adding their coefficients, resulting in \(3w\). This step is vital for simplifying polynomials to their most condensed form and for making the expressions clearer and easier to understand or further manipulate.

Defining Polynomial Expressions

A polynomial expression is like a mathematical phrase composed of variables, coefficients, and exponents connected by addition or subtraction signs. Picture it as a string of beads, where each bead size and color varies, representing different terms within a polynomial.

These expressions are grouped by degrees, determined by the exponent of the variable. In our exercise, the resulting polynomial expression after applying the distributive property and combining like terms is \( -3w^2 + 3w + 60 \). This represents a quadratic polynomial, as the highest degree of the variable \(w\) is 2. It's essential to arrange the terms in descending order of the degree, ensuring that the term with the highest power comes first - organizing thoughts in an essay from most to least important.

Understanding polynomial expressions enables students to tackle a wide range of mathematical problems, from simple algebraic equations to complex calculus functions.