Question

Add, subtract, or multiply, as indicated. Express your answer as a single polynomial in standard form. $$ 6\left(x^{3}+x^{2}-3\right)-4\left(2 x^{3}-3 x^{2}\right) $$

Short answer

-2x^3 + 18x^2 - 18

Step-by-step solution

Distribute the coefficients

Distribute the coefficients 6 and -4 to each term inside the parentheses: 6(x^3 + x^2 - 3) = 6x^3 + 6x^2 - 18 -4(2x^3 - 3x^2) = -8x^3 + 12x^2

Combine like terms

Add or subtract the coefficients of like terms: (6x^3 + 6x^2 - 18) + (-8x^3 + 12x^2) Combine the like terms: (6x^3 - 8x^3) + (6x^2 + 12x^2) - 18 = -2x^3 + 18x^2 - 18

Write the polynomial in standard form

The polynomial is already in standard form when the terms are ordered by decreasing powers of x: -2x^3 + 18x^2 - 18

Key concepts

Polynomial Addition

Polynomial addition involves combining two or more polynomials to create a single polynomial. When you perform polynomial addition, you add the coefficients of like terms together. Like terms are terms that have the same variable raised to the same power. For example, in the polynomials \(6x^2\) and \(4x^2\), the like terms are \(x^2\). Hence, you add 6 and 4 to get \(10x^2\). Here are helpful steps for polynomial addition:

• Identify like terms.
• Add the coefficients of like terms.
• Keep the variables and exponents the same.

This will allow you to simplify the expressions into a single polynomial, as shown in step 2 of our exercise.

Polynomial Subtraction

Polynomial subtraction, similar to addition, involves combining polynomials, but you are subtracting the coefficients of like terms. Let's consider our example from the exercise: when we expand \( -4(2x^3 - 3x^2)\), you subtract \(4 \times 2x^3\) and \(4 \times -3x^2\). This results in \( -8x^3 + 12x^2\). Here’s how to do polynomial subtraction effectively:

• Distribute the subtraction sign through the polynomial.
• Identify like terms.
• Subtract the coefficients of like terms.
• Keep the variables and exponents the same.

Subtracting polynomials can easily be managed just by careful subtraction of like terms.

Standard Form

A polynomial is said to be in standard form when its terms are ordered by descending powers of the variable. For instance, a polynomial in x will have terms ordered by powers of x, starting from the highest power down to the lowest power or constant term, such as \( -2x^3 + 18x^2 - 18\). To convert any polynomial into standard form:

• Expand and simplify the polynomial.
• Combine like terms.
• Order the terms by the degree of the exponents, with the highest degree first.

Standard form makes polynomials easier to read and is essential for performing operations or solving polynomial equations accurately.