Adding, subtracting, and simplifying polynomials are important skills in algebra. This page provides Grade 7 math multiple-choice questions on these topics, accompanied by detailed, step-by-step solutions to help students, parents, and teachers.
To add polynomials in algebra, we group like terms and simplify.
Given: \( (2x + 5) + (4x + 6) \)
1. Group like terms inside parentheses: \( = (2x + 4x) + (5 + 6) \)
2. Simplify: \( = 6x + 11 \)
Given: \( (x^{2} - 6x - 9) + (-5x^{2} + 9x + 2) \)
1. Group like terms inside parentheses: \( = (x^{2} - 5x^{2}) + (-6x + 9x) + (-9 + 2) \)
2. Simplify: \( = -4x^{2} + 3x - 7 \)
When subtracting a second polynomial from a first polynomial, follow these steps:
Given: \( (2xy + x + 5) - (3xy - 2x + 7) \)
1. Change the subtraction into addition and change signs of the terms in the second polynomial: \( = (2xy + x + 5) + (-3xy + 2x - 7) \)
2. Group like terms: \( = (2xy - 3xy) + (x + 2x) + (5 - 7) \)
3. Simplify: \( = -xy + 3x - 2 \)
Add the polynomials: \( (9x - 6) + (-5x + 7) \)
Group the like terms (the \(x\) terms together, and the constants together):
\( = (9x - 5x) + (-6 + 7) \)
\( = 4x + 1 \)
Correct Answer: C
Subtract the polynomials: \( (9x - 6) - (-5x + 7) \)
Change the subtraction to addition by flipping the signs of the second polynomial:
\( = (9x - 6) + (5x - 7) \)
Group like terms:
\( = (9x + 5x) + (-6 - 7) \)
\( = 14x - 13 \)
Correct Answer: A
Add the polynomials: \( (-x^{2} + 5x + 2) + (6x^{2} + x) \)
Group the like terms (the \(x^2\) terms, the \(x\) terms, and the constants):
\( = (-x^{2} + 6x^{2}) + (5x + x) + 2 \)
\( = 5x^{2} + 6x + 2 \)
Correct Answer: B
Subtract the polynomials: \( (-x^{2} + 5x) - (6x^{2} + x - 2) \)
Distribute the negative sign to the second polynomial:
\( = (-x^{2} + 5x) + (-6x^{2} - x + 2) \)
Group like terms:
\( = (-x^{2} - 6x^{2}) + (5x - x) + 2 \)
\( = -7x^{2} + 4x + 2 \)
Correct Answer: D
Add the polynomials: \( (-7x^{2}y + xy + 3x + 2) + (5x^{2}y - 5xy - 6x - 7) \)
Group the like terms:
\( = (-7x^{2}y + 5x^{2}y) + (xy - 5xy) + (3x - 6x) + (2 - 7) \)
\( = -2x^{2}y - 4xy - 3x - 5 \)
Correct Answer: A
Subtract the polynomials: \( (-7x^{2}y + xy + 3x + 2) - (5x^{2}y - 5xy - 6x - 7) \)
Distribute the negative sign to the second polynomial:
\( = (-7x^{2}y + xy + 3x + 2) + (-5x^{2}y + 5xy + 6x + 7) \)
Group like terms:
\( = (-7x^{2}y - 5x^{2}y) + (xy + 5xy) + (3x + 6x) + (2 + 7) \)
\( = -12x^{2}y + 6xy + 9x + 9 \)
Correct Answer: B
Add the polynomials: \( (xy + 3x + 2) + (3x^{2} + y) \)
Identify like terms. In this case, there are no like terms to combine.
Write all the terms out together, typically ordering from highest degree to lowest:
\( = 3x^{2} + xy + 3x + y + 2 \)
Correct Answer: C
Subtract the polynomials: \( (xy + 3x + 2) - (3x^{2} + y) \)
Distribute the negative sign to the second polynomial:
\( = (xy + 3x + 2) + (-3x^{2} - y) \)
There are no like terms to combine. Write the final expression:
\( = -3x^{2} + xy + 3x - y + 2 \)
Correct Answer: D
Add the polynomials: \( (x^{2}y + 3x + 2) + (2 + 2xy^{2} + 3x) \)
Carefully identify like terms. Note that \(x^2y\) and \(2xy^2\) are NOT like terms because their exponents on the variables are different.
Group the true like terms (\(3x\) with \(3x\), and \(2\) with \(2\)):
\( = x^{2}y + 2xy^{2} + (3x + 3x) + (2 + 2) \)
\( = x^{2}y + 2xy^{2} + 6x + 4 \)
Correct Answer: A
Subtract the polynomials: \( (x^{2}y + 3x + 2) - (2 + xy^{2} + 3x) \)
Distribute the negative sign to the second polynomial:
\( = (x^{2}y + 3x + 2) + (-2 - xy^{2} - 3x) \)
Group like terms. Again, note that \(x^2y\) and \(-xy^2\) are not like terms.
\( = x^{2}y - xy^{2} + (3x - 3x) + (2 - 2) \)
The \(x\) terms and constants cancel out (become zero):
\( = x^{2}y - xy^{2} \)
Correct Answer: C