Definition: A polynomial is an expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Adding or Subtracting Polynomials: like terms are grouped and combined.
For example: (xy + 2x² + 4x + y) + (x² – 6x + 2y – xy) simplifies to:
2x² + x² + 4x – 6x + 2y + y + xy – xy
= 3x² – 2x + 3y
Multiplying Polynomials: use the distributive property and combine like terms. The FOIL method can be used when finding the product of two binomials.
For example: (4x + 2y)(3x² + 4y – 7)
= (4x * 3x²) + (4x * 4y) + (4x * -7) + (2y * 3x²) + (2y * 4y) + (2y * -7)
= 12x³ + 16xy – 28x + 6x²y + 8y² – 14y
= 12x³ + 6x²y + 16xy – 28x + 8y² – 14y
Dividing Polynomials: use long division , ensuring any gaps in degrees are filled with zero placeholders. Remainders are included in the final answer as a numerator over the divisor.
Example: Divide the following polynomials: (7x² + x – 8) ÷ (x – 1)
Step 1: Set up the long division problem just like you would for a basic division problem:
Step 2: Divide the first term in the dividend (7x²) by the first term in the divisor (x).
7x²/ x = 7x
This gives us the first term of the quotient, 7x, which is written above the line.
Multiply 7x by the divisor (x – 1): 7x (x – 1) = 7x² – 7x
Subtract this product from the dividend (7x² + x – 8):
(7x² + x – 8) – (7x² – 7x) = 7x² + x – 8 – 7x² + 7x = 8x – 8
Step 3: Divide the first term in the remaining dividend (8x ) by the first term of the divisor (x) to find the second term of the quotient
8x / x = 8
This gives us the second term of the quotient, 8, which is also written above the line.
Multiply 8 by the divisor (x – 1): 8 (x – 1) = 8x – 8
Subtract this product from the remaining dividend (8x – 8):
(8x – 8) – (8x – 8) = 0, which tells us there is no remainder.
(7x² + x – 8) ÷ (x – 1) = 7x + 8
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