Lesson 2 · from §21 of the 1910 text
Subtracting Polynomials
Rule. Write the subtrahend under the minuend with similar terms in the same column. Change the sign of each term of the subtrahend. Then add each column and write the results in order with their proper signs.
Subtract 5x - 2y - 7z + 2 from 3x + 8y - 5z, checking with x = y = z = 1:
\begin{array}{rrrrcl} 3x & +\,8y & -\,5z & & \to & 3 + 8 - 5 = 6 \\ 5x & -\,2y & -\,7z & +\,2 & \to & 5 - 2 - 7 + 2 = -2 \\ \hline -2x & +\,10y & +\,2z & -\,2 & \to & -2 + 10 + 2 - 2 = 8 \end{array}
And 6 - (-2) = 8 too, so it checks. Another good check is difference + subtrahend = minuend.
Quick check
Subtract 2a - 3b from 5a + b.
Two kinds of "find the expression" questions
- What added to A gives B? The answer is B - A.
- What subtracted from A gives B? The answer is A - B.
Quick check
What must be added to x + 4 to give 3x - 1?
Removing parentheses
A plus sign before a parenthesis leaves the signs inside alone: (4x - 3y) + (3x + 5y) = 7x + 2y. A minus sign before it changes every sign inside: 7c - (4c - 9e) = 3c + 9e.
Quick check
Simplify (6m - 2) - (m - 7).
Exercises
How to type math
Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.
Subtract both ways.
Subtract the first number from the second, and the second from the first: a + 2 and a + 3.
Subtract the first number from the second, and the second from the first: a - 4 and a - 2.
Subtract the first number from the second, and the second from the first: 2 a - 3 and 3 a + 5.
Subtract the first number from the second, and the second from the first: 3 a + 7 and 9 - 4 a.
Subtract the first number from the second, and the second from the first: 4 a and 4 a - 3.
Subtract the first number from the second, and the second from the first: 2 a b - 5 c and 5 c.
Subtract the first number from the second, and the second from the first: 3 x y - a and 5 x y.
Subtract the first number from the second, and the second from the first: 0 and x + 3.
Subtract the first number from the second, and the second from the first: 2 x - 3 and 0.
Subtract the first polynomial from the second. Check by letting each letter equal 1.
Subtract x - 2 y + 3 z from 2 x - 2 y + z.
Subtract 4 x - 8 y - 2 z from 4 x - 5 y + 3 z.
Subtract 3 a - 2 b from 4 a - b + 2 c.
Subtract 5 a - 4 b - 3 c from 3 a - 5 b.
Subtract a - x - y from b - x + y.
Subtract 3 a - 2 b - c + 6 from 4 a - b + 5 c.
Subtract 2 a - 2 b - 2 c + 4 from a - 3 b.
Subtract a + b - c from c - d + e.
Find the expression which added to the first gives the second.
What must be added to x - 2 y + z to give 2 x + 5 y - 3 z?
What must be added to 7 x - 9 y + 3 z to give 5 x - 2 y - 4 z?
What must be added to 5 a - 4 b - 6 c to give 6 a + 3 b?
What must be added to a + b - 2 c to give 5?
What must be added to 3 a b + c to give 3 x y + z?
What must be added to 2 x - 4 y - z to give 0?
What must be added to 2 x^{2} y - 3 x y^{2} to give y^{2} x + y x^{2} + z?
Find the expression which subtracted from the first gives the second.
What must be subtracted from x - 2 y + z to give 3 x + 2 y - z?
What must be subtracted from x^{2} - 7 x + 10 to give 14 x - 8 + 3 x^{2}?
What must be subtracted from x - y + z to give 5 x + 3 y - 8 z?
What must be subtracted from 4 - 8 x^{2} to give x^{2} - 5 x + 6?
What must be subtracted from 3 a - 5 b + c to give 0?
What must be subtracted from 4 a - 6 b + 8 y to give x - 12?
Combined problems.
Subtract the sum of a^2 - 2ab + b^2 and a^2 - 12ab + 20 from a^2 - 13a + 30.
Subtract the sum of a - 3b + c and 4a + 5b - 6c + 4 from a - b + c - x.
From the sum of 5x + 3x^2y - 15xy^2 and -6x - 12y^2x + 7yx^2 subtract 11x - 5x^2y + 7y^2x.
From the sum of 4abc^2 - 3ab^2c + 2a^2bc and 6ac^2b - 5acb^2 - 4a^2b^2c subtract 2c^2ab - 3a^2bc + 7acb^2.
From the sum of 3x - 4xy - 2z and 7xy - 4z - 3x take the sum of 5z - 2xy - a^2bc and 9x - 6ba^2c - z.
Simplify. A minus sign in front of a parenthesis changes the sign of everything inside.
Simplify 4 x - 3 y + 6 + 3 x + 5 y - 10.
Simplify 7 c + 5 d - e - \left(4 c + 5 d - 9 e\right).
Simplify x^{2} + 2 x + 5 + 2 x^{2} + x - 10 - \left(x^{2} - 5 x + 3\right).
Simplify x + 3 y - 2 z + 4 x - 5 y + 3 z - \left(3 x - 2 y - 6 z\right).
Simplify 5 x + 3 y - z + 4 y + 7 z - \left(x - y + 3 z\right).
Simplify 4 x - 3 y + 7 - \left(2 x - 5 y - 4\right) + 4 x - 8.
Simplify 3 c - 5 d - e - \left(5 + 6 d + 11 e\right) - \left(5 c + 4 e\right).
Simplify 3 a + 3 b - 4 c - \left(- 3 b - 3 c - 4\right) - \left(4 a + x - 8 c\right).