Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ 7\left(y^{2}-5 y+3\right)-4\left(3-y^{2}\right) $$

Short answer

11y^2 - 35y + 9

Step-by-step solution

Distribute the Constants

First, distribute the constants in front of the parentheses. For the term with 7, multiply 7 by each term inside the parentheses: 7(y^2 - 5y + 3) = 7y^2 - 35y + 21. For the term with -4, multiply -4 by each term inside the parentheses: -4(3 - y^2) = -12 + 4y^2.

Combine Like Terms

Next, combine the like terms from both expressions obtained after distribution: 7y^2 - 35y + 21 - 12 + 4y^2. Combine the y^2 terms, y terms, and the constants separately.

Simplify the Expression

Sum the like terms to get a single polynomial: (7y^2 + 4y^2) + (-35y) + (21 - 12)This simplifies to 11y^2 - 35y + 9.

Key concepts

polynomial operations

Polynomial operations are essential for working with algebraic expressions. These operations include addition, subtraction, and multiplication. Each of these operations follows specific rules for combining and manipulating terms. For example, when adding or subtracting polynomials, you combine like terms to simplify the expression. When multiplying, you use the distributive property to expand the expressions. Understanding these operations helps in solving and simplifying complex algebraic expressions.

combining like terms

Combining like terms is a crucial step in simplifying polynomials. Like terms have the same variable raised to the same power. For example, in the expression \(7y^2 - 35y + 21 - 12 + 4y^2\), the like terms are \(7y^2\) and \(4y^2\) because they both have the variable \(y^2\).
To combine them, simply add their coefficients: \(7y^2 + 4y^2 = 11y^2\).
Similarly, you handle the constants and linear terms separately:

• Combine constants like 21 and -12 to get 9.
• Combine linear terms like -35y with any other in the same form if present.

This process reduces the expression to a simpler polynomial.

distributive property

The distributive property is used to multiply a single term by each term inside a parenthesis. It states that \(a(b + c) = ab + ac\).
In our example, we multiply 7 by each term in \( (y^2 - 5y + 3) \) and -4 by each term in \( (3 - y^2) \):

\[7(y^2 - 5y + 3) = 7y^2 - 35y + 21\]

\[-4(3 - y^2) = -12 + 4y^2\]

By using the distributive property, we can then combine these expanded expressions, ensuring that each term is properly accounted for before simplifying the polynomial into its final form.