Question: PRACTICE SET 1
Basic Mathematical Operations
Name: ________________________________ Date:
Code No.: _____________________________ Score:
Program: ______________________________
A. Perform the indicated operations by combining the similar terms.
1) $(2x^3 - 3x^2 + 4x) + (-3x^3 + 5x^2 - 2x)$
2) $(3x + 2)(2x^2 - 5xy + 4y^2)$
3) $\frac{2a^3b^2c}{3a^2bc^3}$
4) $\frac{3x^2 - x - 2}{2x - 2}$
5) $\frac{(a^2 - 4)(a + 2)}{a - 2}$
bottom-left: a visible label "B. Simplify by removing symbols of grouping and combine similar terms." appears near the lower-left edge of the page.
User: what type of lessons are there in every problem, and explain it to me and teach how to solve
1. **Problem 1: Add polynomials by combining like terms.**
Given: $$(2x^3 - 3x^2 + 4x) + (-3x^3 + 5x^2 - 2x)$$
- Step 1: Remove parentheses and group like terms:
$$2x^3 - 3x^2 + 4x - 3x^3 + 5x^2 - 2x$$
- Step 2: Combine like terms:
$$ (2x^3 - 3x^3) + (-3x^2 + 5x^2) + (4x - 2x) = -x^3 + 2x^2 + 2x$$
**Lesson:** Adding polynomials involves combining terms with the same variable and exponent.
2. **Problem 2: Multiply binomial and trinomial (polynomial multiplication).**
Given: $$(3x + 2)(2x^2 - 5xy + 4y^2)$$
- Step 1: Use distributive property (FOIL for polynomials):
$$3x(2x^2 - 5xy + 4y^2) + 2(2x^2 - 5xy + 4y^2)$$
- Step 2: Multiply each term:
$$6x^3 - 15x^2y + 12xy^2 + 4x^2 - 10xy + 8y^2$$
- Step 3: Combine like terms if any (none here).
**Lesson:** Multiply each term in the first polynomial by each term in the second.
3. **Problem 3: Simplify a rational expression with variables and exponents.**
Given: $$\frac{2a^3b^2c}{3a^2bc^3}$$
- Step 1: Divide coefficients:
$$\frac{2}{3}$$
- Step 2: Subtract exponents of like bases using $a^m / a^n = a^{m-n}$:
$$a^{3-2} = a^1 = a$$
$$b^{2-1} = b^1 = b$$
$$c^{1-3} = c^{-2} = \frac{1}{c^2}$$
- Step 3: Combine all:
$$\frac{2}{3} \times a \times b \times \frac{1}{c^2} = \frac{2ab}{3c^2}$$
**Lesson:** Simplify fractions by dividing coefficients and subtracting exponents of like variables.
4. **Problem 4: Simplify a rational expression (polynomial division).**
Given: $$\frac{3x^2 - x - 2}{2x - 2}$$
- Step 1: Factor numerator and denominator if possible.
Numerator: $3x^2 - x - 2 = (3x + 2)(x - 1)$
Denominator: $2x - 2 = 2(x - 1)$
- Step 2: Write expression:
$$\frac{(3x + 2)(x - 1)}{2(x - 1)}$$
- Step 3: Cancel common factor $(x - 1)$:
$$\frac{\cancel{(x - 1)}(3x + 2)}{2\cancel{(x - 1)}} = \frac{3x + 2}{2}$$
**Lesson:** Factor numerator and denominator, then cancel common factors to simplify.
5. **Problem 5: Simplify a rational expression involving products and factoring.**
Given: $$\frac{(a^2 - 4)(a + 2)}{a - 2}$$
- Step 1: Factor $a^2 - 4$ as difference of squares:
$$a^2 - 4 = (a - 2)(a + 2)$$
- Step 2: Substitute:
$$\frac{(a - 2)(a + 2)(a + 2)}{a - 2}$$
- Step 3: Cancel common factor $(a - 2)$:
$$\frac{\cancel{(a - 2)}(a + 2)(a + 2)}{\cancel{a - 2}} = (a + 2)^2$$
**Lesson:** Use factoring and cancel common factors to simplify expressions.
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**Summary of lessons:**
- Problem 1: Adding polynomials by combining like terms.
- Problem 2: Multiplying polynomials using distributive property.
- Problem 3: Simplifying rational expressions by dividing coefficients and subtracting exponents.
- Problem 4: Simplifying rational expressions by factoring and canceling common factors.
- Problem 5: Factoring expressions and simplifying by canceling common factors.
Each problem teaches fundamental algebraic manipulation skills essential for simplifying and solving expressions.