Before you read

Answer first — see what you already know.

The total area of two rectangular plots is $(4 x^{2} + 3 x) + (2 x^{2} - 5 x + 1)$ square meters. What is the simplified expression for the total area?

A
$6 x^{2} - 2 x + 1$
B
$6 x^{2} + 8 x + 1$
C
$6 x^{4} - 2 x + 1$
D
$2 x^{2} + 8 x + 1$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

An engineer calculates a stress difference as $(7 x^{2} - 2 x + 5) - (3 x^{2} + 4 x - 1)$ . What is the simplified result?

A
$4 x^{2} + 2 x + 4$
B
$4 x^{2} - 6 x + 6$
C
$10 x^{2} - 6 x + 4$
D
$4 x^{2} - 6 x + 4$

Check your answer

Answer:

Theory

Polynomial Basics

A polynomial is an expression with variables and coefficients, using only addition, subtraction, multiplication, and non-negative integer exponents.

Vocabulary:
- Term: a piece of the polynomial (e.g., $3x^2$ )
- Coefficient: the number in front (e.g., $3$ in $3x^2$ )
- Degree of a term: the exponent on the variable (e.g., $2$ in $3x^2$ )
- Degree of the polynomial: the highest degree among all terms
- Leading coefficient: the coefficient of the highest-degree term
- Standard form: terms ordered from highest to lowest degree

Example: $4x^3 - 2x^2 + 7x - 5$ has degree $3$ , leading coefficient $4$ .

$(2x^2 + 3x) + (x^2 - 5x + 1)$
$3x^2 - 2x + 1$ combine like terms

Adding polynomials

Theory

Adding and Subtracting Polynomials

Combine like terms — terms with the same variable and exponent.

Adding: just combine like terms.
Subtracting: distribute the negative, then combine.

Example 1

$(3x^2 + 5x - 2) + (x^2 - 3x + 7)$

Combine $x^2$ terms: $3x^2 + x^2 = 4x^2$

Combine $x$ terms: $5x + (-3x) = 2x$

Combine constants: $-2 + 7 = 5$

Result: $4x^2 + 2x + 5$

4x^2 + 2x + 5

Example 2

$(5x^2 + 3x - 1) - (2x^2 - 4x + 6)$

Distribute the negative: $5x^2 + 3x - 1 - 2x^2 + 4x - 6$

Combine: $(5-2)x^2 + (3+4)x + (-1-6)$

Result: $3x^2 + 7x - 7$

3x^2 + 7x - 7

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