Algebra of polynomials - Multiplication

Question 1: Product of a Polynomial and its Slope (from file image_d19339.png)

The Question: If the polynomial $r(x)$ is the product of a degree two polynomial $p(x) = -8x^2 - 12x + 2$ and its slope then the coefficient of term with highest power is __________.

Core Concept: The term with the highest power in the product polynomial, $r(x)$, is the product of the highest power terms of the original polynomials.

Detailed Solution:

1. Identify the first polynomial, $p(x)$:

   • $p(x) = -8x^2 - 12x + 2$.
   • Its highest power term is $-8x^2$. Its degree is 2.

2. Find the slope of $p(x)$ by finding its derivative, $p’(x)$:

   • The derivative of $-8x^2$ is $(2 \cdot -8)x^{2-1} = -16x$.
   • The derivative of $-12x$ is $-12$.
   • The derivative of $2$ is $0$.
   • So, the slope is $p’(x) = -16x - 12$.
   • Its highest power term is $-16x$. Its degree is 1.

3. Find the highest power term of the product, $r(x)$:

   • $r(x) = p(x) \times p’(x)$.
   • The highest power term of $r(x)$ is the product of the highest power terms of $p(x)$ and $p’(x)$.
   • Highest Term = $(-8x^2) \times (-16x) = 128x^3$.

4. Identify the coefficient:

   • The coefficient of the highest power term ($x^3$) is 128.

Final Answer: 128

Question 2: Product of Two Polynomials (from file image_d19339.png)

The Question: The product of the polynomials $p(x) = 2x^2 + 2x + 2$ and $q(x) = x^2 - x - 1$ is __________.

Core Concept: Use the distributive property to multiply every term in $p(x)$ by every term in $q(x)$.

Detailed Solution:

1. Set up the multiplication: $(2x^2 + 2x + 2)(x^2 - x - 1)$

2. Distribute each term of the first polynomial:

   • Multiply by $2x^2$: $2x^2(x^2 - x - 1) = 2x^4 - 2x^3 - 2x^2$
   • Multiply by $2x$: $2x(x^2 - x - 1) = 2x^3 - 2x^2 - 2x$
   • Multiply by $2$: $2(x^2 - x - 1) = 2x^2 - 2x - 2$

3. Add the results and combine like terms:

   $$(2x^4 - 2x^3 - 2x^2) + (2x^3 - 2x^2 - 2x) + (2x^2 - 2x - 2)$$
   $$= 2x^4 + (-2x^3 + 2x^3) + (-2x^2 - 2x^2 + 2x^2) + (-2x - 2x) - 2$$
   $$= 2x^4 + 0x^3 - 2x^2 - 4x - 2$$

Final Answer: $2x^4 - 2x^2 - 4x - 2$

Question 3: Multiplication Formula in Summation Notation (from file image_d19281.png)

The Question: Choose the correct option(s) with respect to the product of the polynomials $p(x) = \sum_{k=0}^{n} a_k x^k$ and $q(x) = \sum_{j=0}^{m} b_j x^j$.

Core Concept: The Cauchy Product for Polynomials

This formula looks complex, but it’s a formal way of saying what we do when we multiply. The coefficient of a term like $x^k$ in the final product is found by summing up all the products of coefficients ($a_j b_i$) where the powers add up to $k$ (i.e., $j+i=k$).

The formula is:

Detailed Solution:

1. Degree of the Product: The highest power in the result will be $x^{m+n}$. So the outer sum must go from $k=0$ to $m+n$.
2. Coefficient of $x^k$: As stated in the concept, the coefficient for any term $x^k$ is the sum of products of coefficients $a_j$ and $b_{k-j}$ for all possible $j$ from 0 to $k$. This is written as $\sum_{j=0}^{k} a_j b_{k-j}$.
3. Putting it together: This gives the formula $p(x) \times q(x) = \sum_{k=0}^{m+n} \left( \sum_{j=0}^{k} a_j b_{k-j} \right) x^k$.
4. Commutativity: Since polynomial multiplication is commutative, $p(x) \times q(x) = q(x) \times p(x)$. This means the same formula holds for the product in the reverse order.

Let’s check the options:

• The third option matches the formula for $p(x) \times q(x)$ exactly. TRUE.
• The fourth option gives the same correct formula for $q(x) \times p(x)$. TRUE.
• The other options incorrectly alter the exponents or the indices of the sums.

Final Answer: The correct options are:

• $p(x) \times q(x) = \sum_{k=0}^{m+n} \sum_{j=0}^{k} (a_j b_{k-j}) x^k$
• $q(x) \times p(x) = \sum_{k=0}^{m+n} \sum_{j=0}^{k} (a_j b_{k-j}) x^k$

Question 4: Analyzing the Product of Polynomials (from file image_d18f94.png)

The Question: If $r(x) = p(x)q(x)$ where $p(x) = -2x^9 + 4x^5 + 2x^2 + 1$ and $q(x) = 8x^4 - 2x^2 + x$ then choose the correct option(s) from the following.

Detailed Solution:

Let’s evaluate each statement.

• “Degree of $r(x)=13$ and coefficient of $x^{13}=-16$”

  • Degree of $r(x) = \text{deg}(p) + \text{deg}(q) = 9 + 4 = 13$.
  • The coefficient of the highest power term ($x^{13}$) is the product of the leading coefficients of $p(x)$ and $q(x)$.
  • Coefficient = $(-2) \times (8) = -16$.
  • This statement is TRUE.

• “Degree of polynomial $r(x)=36$ and it has 0 coefficient.”

  • FALSE. The degree is 13, not 36 ($9 \times 4$).

• “Coefficient of $x^2$ in $r(x)$ and $q(x)$ is -2”

  • In $q(x) = 8x^4 - 2x^2 + x$, the coefficient of $x^2$ is indeed -2.
  • To get the $x^2$ term in $r(x)$, we find all pairs of terms from $p(x)$ and $q(x)$ that multiply to give a power of 2:
    • $(2x^2 \text{ from } p) \times (\text{constant term from } q) = 2x^2 \times 0 = 0$
    • $(\text{constant term from } p) \times (-2x^2 \text{ from } q) = 1 \times (-2x^2) = -2x^2$
  • The total coefficient of $x^2$ in $r(x)$ is $0 + (-2) = -2$.
  • The statement is TRUE.

• “Coefficient of $x^8$ in $r(x)$ is 0”

  • To get an $x^8$ term, we need terms with powers that sum to 8. Let’s check the possibilities:
    • $p(x)$ has powers 9, 5, 2, 0.
    • $q(x)$ has powers 4, 2, 1.
    • Can we combine any pair to get 8? (9+?) no, (5+?) no, (2+?) no, (0+?) no.
  • Since there is no way to form an $x^8$ term, its coefficient must be 0. This statement is TRUE.

• “Coefficient of $x^9$ in $r(x)$ and $p(x)$ is 32 and -2 respectively”

  • The coefficient of $x^9$ in $p(x)$ is clearly -2.
  • To get the $x^9$ term in $r(x)$, we find all pairs with powers that sum to 9:
    • $(-2x^9 \text{ from } p) \times (\text{constant from } q) = -2x^9 \times 0 = 0$
    • $(4x^5 \text{ from } p) \times (8x^4 \text{ from } q) = 32x^9$
  • The total coefficient of $x^9$ in $r(x)$ is $0 + 32 = 32$.
  • The statement is TRUE.

Final Answer: The correct statements are:

• Degree of $r(x) = 13$ and coefficient of $x^{13} = -16$
• Coefficient of $x^2$ in $r(x)$ and $q(x)$ is -2
• Coefficient of $x^8$ in $r(x)$ is 0
• Coefficient of $x^9$ in $r(x)$ and $p(x)$ is 32 and -2 respectively