Question

ind the sum and difference of the following polynomials and simplify the results:
7 ((4m^2 - 3m^3 + 2m) + (m^3 - m))
8 ((5x^4 - 3y^2 + 6x) - (3x^4 - 7y^2 - 6x))
(note: the left - side and upper - side text are partially displayed and not fully related to the main polynomial operation problems here, so only the clear polynomial operation problems are retained.)

Explanation:

Problem 7: Sum and Difference of Polynomials

Step 1: Find the Sum

Add the two polynomials:
$$(4m^2 - 3m^3 + 2m) + (m^3 - m)$$
Combine like terms:

• For \(m^3\): \(-3m^3 + m^3 = -2m^3\)
• For \(m^2\): \(4m^2\) (no other \(m^2\) term)
• For \(m\): \(2m - m = m\)

Result: \(-2m^3 + 4m^2 + m\)

Step 2: Find the Difference

Subtract the second polynomial from the first:
$$(4m^2 - 3m^3 + 2m) - (m^3 - m)$$
Distribute the negative sign:
$$4m^2 - 3m^3 + 2m - m^3 + m$$
Combine like terms:

• For \(m^3\): \(-3m^3 - m^3 = -4m^3\)
• For \(m^2\): \(4m^2\)
• For \(m\): \(2m + m = 3m\)

Result: \(-4m^3 + 4m^2 + 3m\)

Problem 8: Simplify the Polynomial Expression

Step 1: Distribute the Subtraction

$$(5x^4 - 3y^2 + 6x) - (3x^4 - 7y^2 - 6x)$$
$$= 5x^4 - 3y^2 + 6x - 3x^4 + 7y^2 + 6x$$

Step 2: Combine Like Terms

• For \(x^4\): \(5x^4 - 3x^4 = 2x^4\)
• For \(y^2\): \(-3y^2 + 7y^2 = 4y^2\)
• For \(x\): \(6x + 6x = 12x\)

Result: \(2x^4 + 4y^2 + 12x\)

Problem 6: Evaluate the Expression

The expression is \(\frac{1}{2}ab^3 - 4ab\) (assuming a typo, as "2 - ab³ - 4ab" is unclear; corrected to \(\frac{1}{2}ab^3 - 4ab\) for clarity).

Given \(a = 4\), \(b = -2\):

Step 1: Calculate \(ab^3\)

\(ab^3 = 4 \times (-2)^3 = 4 \times (-8) = -32\)

Step 2: Calculate \(\frac{1}{2}ab^3\)

\(\frac{1}{2} \times (-32) = -16\)

Step 3: Calculate \(4ab\)

\(4 \times 4 \times (-2) = -32\)

Step 4: Subtract

\(-16 - (-32) = -16 + 32 = 16\)

Final Answers:

• Problem 7 (Sum): \(\boldsymbol{-2m^3 + 4m^2 + m}\)
• Problem 7 (Difference): \(\boldsymbol{-4m^3 + 4m^2 + 3m}\)
• Problem 8: \(\boldsymbol{2x^4 + 4y^2 + 12x}\)
• Problem 6 (Evaluated): \(\boldsymbol{16}\) (assuming the corrected expression)

Answer:

The expression is \(\frac{1}{2}ab^3 - 4ab\) (assuming a typo, as "2 - ab³ - 4ab" is unclear; corrected to \(\frac{1}{2}ab^3 - 4ab\) for clarity).

Given \(a = 4\), \(b = -2\):

Step 1: Calculate \(ab^3\)

\(ab^3 = 4 \times (-2)^3 = 4 \times (-8) = -32\)

Step 2: Calculate \(\frac{1}{2}ab^3\)

\(\frac{1}{2} \times (-32) = -16\)

Step 3: Calculate \(4ab\)

\(4 \times 4 \times (-2) = -32\)

Step 4: Subtract

\(-16 - (-32) = -16 + 32 = 16\)

Final Answers:

• Problem 7 (Sum): \(\boldsymbol{-2m^3 + 4m^2 + m}\)
• Problem 7 (Difference): \(\boldsymbol{-4m^3 + 4m^2 + 3m}\)
• Problem 8: \(\boldsymbol{2x^4 + 4y^2 + 12x}\)
• Problem 6 (Evaluated): \(\boldsymbol{16}\) (assuming the corrected expression)