Problem 98 - College Algebra

Question

Explain how to subtract polynomials.

Step-by-Step Solution

Answer

To subtract polynomials, distribute any negative sign present before a polynomial, line up like terms of the polynomials, perform subtraction term by term, and combine any remaining like terms.

Step 1: Distribute any Negative Sign

If a negative sign is right before a polynomial, distribute it to all the terms inside the polynomial. This changes the sign of all the terms inside the brackets. For instance, \( -(2x^2 + 5x - 3) \) changes to \( -2x^2 - 5x + 3 \). Doing this helps when subtracting the polynomials later.

Step 2: Line up Like Terms

Align like terms under one another. Like terms are terms with the same variable(s) and exponent(s). For instance, if you're subtracting \( 4x^3 - 2x^2 + 8x - 1 \) from \( 3x^3 + x^2 - 5x +2 \), arrange them like so: \[ \begin{align*} & 4x^3 - 2x^2 + 8x - 1 \ - & 3x^3 + x^2 - 5x + 2 \end{align*} \] This makes it easier to subtract.

Step 3: Perform Subtraction

Subtract the lower polynomial from the upper polynomial term by term. Subtract coefficients of like terms from each other. Continuing from the previous example, \( (4x^3 - 3x^3) - (2x^2 - x^2) + (8x - -5x) - (1 - 2) \) results in: \( x^3 - x^2 - 13x - 3 \)

Step 4: Combine Like Terms

Upon subtracting, if there are like terms, combine them by addition or substraction. In our example, there are no like terms left to combine, so our final polynomial is: \( x^3 - x^2 - 13x - 3 \)