Question 1

How do you add or subtract polynomials?

Accepted Answer

Combine like terms — terms with the same variable and exponent. For addition: (3x² + 2x − 5) + (x² − 4x + 3) = (3+1)x² + (2−4)x + (−5+3) = 4x² − 2x − 2. For subtraction, distribute the negative first: (3x² + 2x − 5) − (x² − 4x + 3) = 3x² + 2x − 5 − x² + 4x − 3 = 2x² + 6x − 8. Like terms must have exactly the same variable(s) and exponent(s).

Question 2

How do you multiply two polynomials?

Accepted Answer

Use the distributive property — multiply every term in the first polynomial by every term in the second, then combine like terms. Example: (x + 3)(x² − 2x + 5) = x³ − 2x² + 5x + 3x² − 6x + 15 = x³ + x² − x + 15. For two binomials, the FOIL shortcut (First, Outer, Inner, Last) gives the same result: (a+b)(c+d) = ac + ad + bc + bd.

Question 3

What is the degree of a polynomial?

Accepted Answer

The degree is the highest exponent of x in the polynomial. Degree 0 = constant (e.g., 7). Degree 1 = linear (e.g., 3x+2). Degree 2 = quadratic (e.g., x²−5x+6). Degree 3 = cubic. Degree 4 = quartic. Degree 5 = quintic. When multiplying polynomials, the degree of the product equals the sum of the degrees: degree 2 × degree 3 = degree 5.

Question 4

What is FOIL and when does it apply?

Accepted Answer

FOIL is a mnemonic for multiplying two binomials: (a+b)(c+d) = ac (First) + ad (Outer) + bc (Inner) + bd (Last). Example: (2x+3)(x−4) = 2x² − 8x + 3x − 12 = 2x² − 5x − 12. FOIL only works for exactly two binomials. For trinomials or larger polynomials, use full distribution — FOIL is just a shortcut for the 2×2 case.

Question 5

How do you find the value of a polynomial at a specific x?

Accepted Answer

Substitute the x value and evaluate using order of operations. Example: P(x) = 2x³ − x + 5 at x = 3: P(3) = 2(27) − 3 + 5 = 54 + 2 = 56. Horner's method is more efficient for large polynomials: rewrite as ((2x − 0)x − 1)x + 5 and evaluate left to right. This reduces multiplications from O(n²) to O(n).

Question 6

What is standard form for a polynomial?

Accepted Answer

Standard form lists terms in descending order of degree: highest power first, down to the constant. Example: x⁴ − 3x³ + 2x² + x − 7. The leading term is the highest-degree term (x⁴), and the leading coefficient is its coefficient (1 in this case). Standard form makes it easy to identify the degree, compare polynomials, and perform polynomial long division.

Operation	Rule	Example	Result
Addition	Combine like terms	(3x²+2x−5)+(x²−4x+3)	4x²−2x−2
Subtraction	Distribute negative, combine	(3x²+2x−5)−(x²−4x+3)	2x²+6x−8
FOIL (binomials)	(a+b)(c+d)=ac+ad+bc+bd	(x+3)(x−2)	x²+x−6
Distribute (trinomial)	Each term × each term	(x+1)(x²+x+1)	x³+2x²+2x+1
Square of binomial	(a+b)²=a²+2ab+b²	(x+4)²	x²+8x+16
Difference of squares	(a+b)(a−b)=a²−b²	(x+5)(x−5)	x²−25

Degree	Name	Standard Form	Example
0	Constant	a	7
1	Linear	ax + b	3x − 2
2	Quadratic	ax² + bx + c	x² − 5x + 6
3	Cubic	ax³ + bx² + cx + d	2x³ − x + 4
4	Quartic	ax⁴ + bx³ + cx² + dx + e	x⁴ − 3x² + 1
5	Quintic	ax⁵ + …	x⁵ − x³ + 2x

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