Math Problems: Degree, Substitution, And Division

Oct 16, 2025 by ADMIN 50 views

Determining the Degree of Polynomial Sums

Alright, guys, let's dive into the fascinating world of polynomials! Polynomials are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Understanding the degree of a polynomial is crucial for various mathematical operations and analyses. The degree of a polynomial is simply the highest power of the variable in the polynomial. For example, in the polynomial $P(x) = x^5 + 3x^2 - 7$ , the degree is 5 because the highest power of x is 5.

When we add two polynomials, the degree of the resulting polynomial is determined by the highest degree term that survives the addition. In other words, if the highest degree terms in the original polynomials do not cancel each other out, then the degree of the sum will be the same as the highest degree among the original polynomials. However, if the highest degree terms do cancel out, then the degree of the sum will be lower.

Now, let’s tackle the problem at hand. We are given two polynomials:

$P(x) = y^3 + 5y^2 + y - 1$ $Q(y) = 4y^3 + 5y^2 + 3y - 7$

To find the degree of $P(x) + Q(x)$ , we first need to add the two polynomials together:

$P(x) + Q(x) = (y^3 + 5y^2 + y - 1) + (4y^3 + 5y^2 + 3y - 7)$

Combine like terms:

$P(x) + Q(x) = (1+4)y^3 + (5+5)y^2 + (1+3)y + (-1-7)$ $P(x) + Q(x) = 5y^3 + 10y^2 + 4y - 8$

The resulting polynomial is $5y^3 + 10y^2 + 4y - 8$ . The highest power of the variable y in this polynomial is 3. Therefore, the degree of $P(x) + Q(x)$ is 3. Understanding polynomial degrees helps in various applications, such as curve fitting, solving equations, and analyzing the behavior of functions. Keep practicing, and you'll become a polynomial pro in no time!

Solving for x Using Substitution

Alright, let's switch gears and talk about solving equations using substitution! Substitution is a powerful technique used to simplify and solve equations by replacing one variable or expression with another. It's particularly useful when dealing with composite functions or systems of equations. The basic idea is to express one variable in terms of another and then substitute that expression into the other equation, effectively reducing the number of variables and making the equation easier to solve.

In this problem, we are given the function:

$F(x) = 2y^3 + 3y - 2$

and we are told that $y = -1$ . Our goal is to find the value of $F(x)$ when $y = -1$ . To do this, we simply substitute -1 for y in the expression for $F(x)$ :

$F(x) = 2(-1)^3 + 3(-1) - 2$

Now, we evaluate the expression:

$F(x) = 2(-1) - 3 - 2$ $F(x) = -2 - 3 - 2$ $F(x) = -7$

So, when $y = -1$ , the value of $F(x)$ is -7. Substitution is a fundamental technique in algebra and calculus. Mastering it will significantly enhance your problem-solving abilities. Whether you're dealing with complex equations or simple expressions, substitution can be your best friend. Keep practicing, and you'll be solving equations like a champ!

Finding the Quotient Through Polynomial Division

Now, let's tackle polynomial division. Polynomial division is similar to long division with numbers, but instead of dividing numbers, we're dividing polynomials. It's a method used to divide a polynomial by another polynomial of lower or equal degree. The result of the division gives us a quotient and a remainder. If the remainder is zero, it means the divisor is a factor of the dividend.

We are given the polynomial:

$P(x) = 3y^4 - 5y^3 + 11y^2 + 6y - 10$

and we want to divide it by $y + 2$ . To find the quotient, we can use synthetic division or long division. Here, I'll demonstrate the process using synthetic division, as it's generally quicker for dividing by a linear factor of the form $y - c$ .

Step 1: Set up the synthetic division.

Write down the coefficients of the polynomial $P(x)$ : 3, -5, 11, 6, -10. Since we are dividing by $y + 2$ , we use -2 as the divisor.

-2 | 3 -5 11 6 -10

Step 2: Perform the synthetic division.

Bring down the first coefficient (3).

-2 | 3 -5 11 6 -10
  |
  | 3

Multiply the divisor (-2) by the number you brought down (3) and write the result under the next coefficient (-5).

-2 | 3 -5 11 6 -10
  | -6
  | 3

Add the numbers in the second column (-5 and -6) and write the sum below.

-2 | 3 -5 11 6 -10
  | -6
  | 3 -11

Multiply the divisor (-2) by the result (-11) and write it under the next coefficient (11).

-2 | 3 -5 11 6 -10
  | -6 22
  | 3 -11

Add the numbers in the third column (11 and 22) and write the sum below.

-2 | 3 -5 11 6 -10
  | -6 22
  | 3 -11 33

Multiply the divisor (-2) by the result (33) and write it under the next coefficient (6).

-2 | 3 -5 11 6 -10
  | -6 22 -66
  | 3 -11 33

Add the numbers in the fourth column (6 and -66) and write the sum below.

-2 | 3 -5 11 6 -10
  | -6 22 -66
  | 3 -11 33 -60

Multiply the divisor (-2) by the result (-60) and write it under the last coefficient (-10).

-2 | 3 -5 11 6 -10
  | -6 22 -66 120
  | 3 -11 33 -60

Add the numbers in the last column (-10 and 120) and write the sum below.

-2 | 3 -5 11 6 -10
  | -6 22 -66 120
  | 3 -11 33 -60 110

Step 3: Interpret the results.

The numbers in the bottom row, excluding the last one, are the coefficients of the quotient. The last number is the remainder. In this case, the coefficients are 3, -11, 33, and -60. The remainder is 110. Since the original polynomial was of degree 4 and we divided by a polynomial of degree 1, the quotient will be of degree 3.

Therefore, the quotient is:

$3y^3 - 11y^2 + 33y - 60$

And the remainder is 110.

So, when $P(x) = 3y^4 - 5y^3 + 11y^2 + 6y - 10$ is divided by $y + 2$ , the quotient is $3y^3 - 11y^2 + 33y - 60$ .

Polynomial division is a fundamental tool in algebra and calculus. It helps in factoring polynomials, finding roots, and simplifying complex expressions. Keep practicing, and you'll become a polynomial division expert!