Question

lesson 3 - adding and subtracting polynomials
mth1w
jensen
unit 3 - algebraic expressions
ws 12 p.1
part 1: adding polynomials
polynomial: an algebraic expression consisting of one or more terms connected by addition or
subtraction operators.
when adding polynomials, you can simply drop the brackets and collect the like terms.
for example:
(4x + 3) + (7x + 2)
= 4x + 3 + 7x + 2
step 1: remove the brackets
step 2: rearrange like terms into groups
step 3: collect the like terms
example 1: simplify the following expressions
a) (3y + 5) + (7y - 4)
b) (2p - 2) + (4p - 7)
c) (6x - 12) + (-9x - 4) + (x + 14)
d) (5x - 4y - 1) + (-2x + 5y + 13)

Explanation:

) + 14 = -2\)? Wait, maybe I made a mistake. Let's do it again:
\(6x - 9x + x = (6 + 1 - 9)x = (-2)x\)
\(-12 - 4 + 14 = (-16) + 14 = -2\)? Wait, no, \(-12 - 4\) is \(-16\), plus \(14\) is \(-2\). Wait, but let's check again:
Wait, \(6x -12 -9x -4 +x +14\)
\(x\)-terms: \(6x -9x +x = (6 -9 +1)x = (-2)x\)
Constants: \(-12 -4 +14 = (-16) +14 = -2\)? Wait, no, \(-12 +14 = 2\), then \(2 -4 = -2\). Yes. So, \(-2x - 2\)? Wait, no, wait: \(-12 -4 = -16\), \(-16 +14 = -2\). So, \(-2x -2\)? Wait, no, let's add step by step:
\(6x -9x +x = (6 + 1 - 9)x = (-2)x\)
\(-12 -4 +14 = (-12 +14) -4 = 2 -4 = -2\). So, \(-2x - 2\)? Wait, but let's check again:
Original expression: \(6x -12 -9x -4 +x +14\)
Combine \(x\) terms: \(6x -9x +x = (6 -9 +1)x = (-2)x\)
Combine constants: \(-12 -4 +14 = (-16) +14 = -2\). So, \(-2x -2\)? Wait, maybe I messed up the signs. Wait, the second bracket is \(-9x -4\), so when we remove the bracket, it's \(-9x -4\), correct. Third bracket is \(x +14\), so \(+x +14\). So:
\(6x -12 -9x -4 +x +14\)
Now, \(6x -9x +x = (6 +1 -9)x = (-2)x\)
\(-12 -4 +14 = (-12 +14) -4 = 2 -4 = -2\). So, \(-2x -2\)? Wait, but let's check with another approach:
\(6x -9x +x = -2x\)
\(-12 -4 +14 = (-12 +14) + (-4) = 2 -4 = -2\). Yes, so \(-2x -2\)? Wait, but maybe I made a mistake. Wait, let's compute all terms:
\(6x -12 -9x -4 +x +14 = (6x -9x +x) + (-12 -4 +14) = (-2x) + (-2)\)? Wait, no, \(-12 -4 is -16, +14 is -2. So, \(-2x -2\). Wait, but let's check with actual numbers. Let \(x = 1\):
Original: \((6(1) -12) + (-9(1) -4) + (1 +14) = (6 -12) + (-9 -4) + 15 = (-6) + (-13) + 15 = (-19) +15 = -4\)
Our result: \(-2(1) -2 = -2 -2 = -4\). Correct. So, \(-2x -2\) is correct. Wait, but earlier I thought maybe a mistake, but it's correct.

Part d: \((5x - 4y - 1)+(-2x + 5y + 13)\)

Step 1: Remove the Brackets

Drop the brackets: \(5x - 4y - 1 - 2x + 5y + 13\)

Step 2: Rearrange like terms into groups

Group \(x\)-terms, \(y\)-terms, and constants: \((5x - 2x)+(-4y + 5y)+(-1 + 13)\)

Step 3: Collect the like terms

Combine \(x\)-terms: \(5x - 2x = 3x\)
Combine \(y\)-terms: \(-4y + 5y = y\)
Combine constants: \(-1 + 13 = 12\)
So, \(3x + y + 12\) (which matches the handwritten work in the image)

Final Answers:

a) \(\boldsymbol{10y + 1}\)
b) \(\boldsymbol{6p - 9}\)
c) \(\boldsymbol{-2x - 2}\)
d) \(\boldsymbol{3x + y + 12}\)

Answer:

) + 14 = -2\)? Wait, maybe I made a mistake. Let's do it again:
\(6x - 9x + x = (6 + 1 - 9)x = (-2)x\)
\(-12 - 4 + 14 = (-16) + 14 = -2\)? Wait, no, \(-12 - 4\) is \(-16\), plus \(14\) is \(-2\). Wait, but let's check again:
Wait, \(6x -12 -9x -4 +x +14\)
\(x\)-terms: \(6x -9x +x = (6 -9 +1)x = (-2)x\)
Constants: \(-12 -4 +14 = (-16) +14 = -2\)? Wait, no, \(-12 +14 = 2\), then \(2 -4 = -2\). Yes. So, \(-2x - 2\)? Wait, no, wait: \(-12 -4 = -16\), \(-16 +14 = -2\). So, \(-2x -2\)? Wait, no, let's add step by step:
\(6x -9x +x = (6 + 1 - 9)x = (-2)x\)
\(-12 -4 +14 = (-12 +14) -4 = 2 -4 = -2\). So, \(-2x - 2\)? Wait, but let's check again:
Original expression: \(6x -12 -9x -4 +x +14\)
Combine \(x\) terms: \(6x -9x +x = (6 -9 +1)x = (-2)x\)
Combine constants: \(-12 -4 +14 = (-16) +14 = -2\). So, \(-2x -2\)? Wait, maybe I messed up the signs. Wait, the second bracket is \(-9x -4\), so when we remove the bracket, it's \(-9x -4\), correct. Third bracket is \(x +14\), so \(+x +14\). So:
\(6x -12 -9x -4 +x +14\)
Now, \(6x -9x +x = (6 +1 -9)x = (-2)x\)
\(-12 -4 +14 = (-12 +14) -4 = 2 -4 = -2\). So, \(-2x -2\)? Wait, but let's check with another approach:
\(6x -9x +x = -2x\)
\(-12 -4 +14 = (-12 +14) + (-4) = 2 -4 = -2\). Yes, so \(-2x -2\)? Wait, but maybe I made a mistake. Wait, let's compute all terms:
\(6x -12 -9x -4 +x +14 = (6x -9x +x) + (-12 -4 +14) = (-2x) + (-2)\)? Wait, no, \(-12 -4 is -16, +14 is -2. So, \(-2x -2\). Wait, but let's check with actual numbers. Let \(x = 1\):
Original: \((6(1) -12) + (-9(1) -4) + (1 +14) = (6 -12) + (-9 -4) + 15 = (-6) + (-13) + 15 = (-19) +15 = -4\)
Our result: \(-2(1) -2 = -2 -2 = -4\). Correct. So, \(-2x -2\) is correct. Wait, but earlier I thought maybe a mistake, but it's correct.

Part d: \((5x - 4y - 1)+(-2x + 5y + 13)\)

Step 1: Remove the Brackets

Drop the brackets: \(5x - 4y - 1 - 2x + 5y + 13\)

Step 2: Rearrange like terms into groups

Group \(x\)-terms, \(y\)-terms, and constants: \((5x - 2x)+(-4y + 5y)+(-1 + 13)\)

Step 3: Collect the like terms

Combine \(x\)-terms: \(5x - 2x = 3x\)
Combine \(y\)-terms: \(-4y + 5y = y\)
Combine constants: \(-1 + 13 = 12\)
So, \(3x + y + 12\) (which matches the handwritten work in the image)

Final Answers:

a) \(\boldsymbol{10y + 1}\)
b) \(\boldsymbol{6p - 9}\)
c) \(\boldsymbol{-2x - 2}\)
d) \(\boldsymbol{3x + y + 12}\)