2.1E Exercises

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Combining Like Terms

Simplify.

\(3x + 2 - x + 4\)
\(-11x + 3 + 8x - 3\)
\(2a^2b - 3ab + 5a^2b - ab\)
\(4x^2 - 2xy + 3xy - x^2 + 5y\)

Answer

Hint: What does "Simplify" mean when a problem looks like this? You are aiming to combine like terms as much as possible. Go try them and come back.

\( 2x + 6 \)
\( -3x \)
\( 7a^2b - 4ab \)
\( 3x^2 + 3xy + 5y \)

Adding and Subtracting Polynomials

Simplify.

\( (3x^2 - 2x + 5) + (4x^2 + 3x - 1) \)
\( (5a^3 + 2a^2 - 4a) - (a^2 + 6a) \)
\( (4y^2 - 3y + 2) - (2y^2 + 5y - 1) + (y^3 + 3) \)
\( (100x + 1) - (0.5x^2 - 30x) + 0.25x^2 \)

Answer

Hint: What does "Simplify" mean when a problem looks like this? The parentheses should be going away (watch out for subtractions) and like terms should all be combined. Go try them and come back.

\( 7x^2 + x + 4 \)
\( 5a^3 + a^2 - 10a \)
\( y^3 + 2y^2 - 8y +6 \)
\( -0.25 x^2 + 130x + 1 \)

Basic FOIL Practice

Expand and simplify.

\((x + 2)(x + 3)\)
\( (2x - 1)(x+1) \)
\((2y - 5)(3y + 4)\)
\((a - 4)(a + 4)\)
\(\left(3x - \frac{1}{2}\right)\left(x - 1 \right)\)
\( \left(\frac{1}{2}x +1 \right)^2 \)
\((2a + \sqrt{2})(a - 2)\)
\( (e^x + 1)(7x + 2) \)

Answer

Hint: "Expand" is the math word for multiplying everything out, like FOIL. Then "simplify" will mean combining any like terms.

\( x^2 + 5x + 6 \)
\( 2x^2 + x - 1\)
\( 6y^2 -7y - 20 \)
\( a^2 - 4 \)
\( 3x^2 - \frac{7}{2}x + \frac{1}{2} \)
\( \frac{1}{4}x^2 + x + 1 \)
\( 2a^2 - 4a + a \sqrt{2} - 2 \sqrt{2} \)
\( 7xe^x + 2e^x + 7x + 2 \)

Advanced Expanding

Expand and simplify.

\((a + 2)(a^2 - 3a + 1)\)
\((x - 1)(2x^2 + x + 1) \)
\( (a -b - 1)(a^2 + 2ab + b^2) \)
\((4x - 2y + 3)(2x + 3y - 1)\)
\( (x + 3)^3 \)
\( (a - b)^3 \)
\( (x+1)(x-2)^2 \)
\( (1+u)(2+u)^2 \)

Answer

\( a^3 - a^2 - 5a + 2 \)
\( 2x^3 - x^2 - 1\)
\( 8x^2 + 8xy - 6y^2 + 2x + 11y - 3 \)
\( -a^2 + a^3 - 2 a b + a^2 b - b^2 - a b^2 - b^3 \)
\( x^3 + 9x^2 + 27x + 27 \)
\( a^3 - 3 a^2 b + 3 a b^2 - b^3 \)
\( x^3 - 3x^2 + 4 \)
\( 4 + 8 u + 5 u^2 + u^3\)

Combo Simplification

Simplify.

\( 3(x+h) - 3x \)
\( (x+h)^2 - x^2 \)
\( (x - 1)^2 + (y+1)^2 \)
\( \sqrt{x}( \sqrt{x} + 2) - (x^2 + \sqrt{x}) \)
\( (3 + b)(3 - b) - (3+b)(3+b) \)
\( (3x^2)(x^2+1) + (2x)(x^3 - 2) \)

Answer

\( 3h\)
\( 2xh + h^2 \)
\( x^2 - 2x + y^2 + 2y +2 \)
\( x + \sqrt{x} - x^2 \)
\( -2b^2 - 6b \) (Be careful with that subtraction!)
\( -4 x + 3 x^2 + 5 x^4\)

Interpreting Expressions With Parentheses

Identify whether the expressions as presented are algebraically equivalent. That is, by using legal algebraic manipulations, can you make them match? When are parentheses important? Investigate by expanding and simplifying each expression as much as possible.

1.	\( 5x + (x^2 + 4) \)	\( 5x + x^2 + 4 \)	6.	\( (2x^2 + 2) + (x - 1) \)	\( x - 1 + (2x^2 + 2) \)
2.	\( x^2 + (-3) \)	\( x^2 - 3 \)	7.	\( 3 - (a^2 + b^2) \)	\( 3 - a^2 + b^2 \)
3.	\( (6x^2 - 3)-2 \)	\( (6x^2 - 3)(-2) \)	8.	\( 3(x+y)^2 \)	\( (3x + 3y)^2 \)
4.	\( (-2)(6x^2 - 3) \)	\( -2(6x^2 - 3) \)	9.	\( (-2a + b)^2 \)	\( -2(a +b)^2 \)
5.	\( (9)(3)(x+y) \)	\( 9(3)(x+y)\)	10.	\( (x+3)^2 \)	\( (x+3)(x+3) \)

Answer

Equivalent.
Equivalent.
Not equivalent.
Equivalent.
Equivalent.
Equivalent.
Not equivalent.
Not equivalent! Expand the perfect square before distributing the \(3\).
Not equivalent.
Equivalent.

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