Divide The Polynomial By The Binomial:${ \left(y 4-y 2+8y+5\right) \div (y+1) }$

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Introduction

Dividing a polynomial by a binomial is a fundamental concept in algebra, and it's an essential skill to master for solving equations and manipulating expressions. In this article, we'll explore the process of dividing a polynomial by a binomial, using the given example: (y4−y2+8y+5)÷(y+1)\left(y^4-y^2+8y+5\right) \div (y+1). We'll break down the steps involved and provide a clear, step-by-step guide on how to perform this division.

What is Polynomial Division?

Polynomial division is the process of dividing a polynomial by another polynomial, resulting in a quotient and a remainder. In this case, we're dividing a polynomial by a binomial, which is a polynomial with only two terms. The binomial is (y+1)(y+1), and the polynomial is (y4−y2+8y+5)\left(y^4-y^2+8y+5\right).

Step 1: Write Down the Division

To start the division, we write down the dividend (the polynomial being divided) and the divisor (the binomial). We'll use the given example:

(y4−y2+8y+5)÷(y+1)\left(y^4-y^2+8y+5\right) \div (y+1)

Step 2: Divide the Leading Term

The first step in polynomial division is to divide the leading term of the dividend by the leading term of the divisor. In this case, the leading term of the dividend is y4y^4, and the leading term of the divisor is yy. We'll divide y4y^4 by yy to get y3y^3.

y4÷y=y3y^4 \div y = y^3

Step 3: Multiply the Divisor by the Quotient

Next, we'll multiply the divisor (y+1)(y+1) by the quotient y3y^3. This will give us a new polynomial that we'll subtract from the dividend.

(y+1)×y3=y4+y3(y+1) \times y^3 = y^4 + y^3

Step 4: Subtract the Product from the Dividend

Now, we'll subtract the product (y4+y3)(y^4 + y^3) from the dividend (y4−y2+8y+5)\left(y^4-y^2+8y+5\right). This will give us a new polynomial that we'll use in the next step.

(y4−y2+8y+5)−(y4+y3)=−y2+8y+5−y3\left(y^4-y^2+8y+5\right) - (y^4 + y^3) = -y^2 + 8y + 5 - y^3

Step 5: Bring Down the Next Term

We'll bring down the next term from the dividend, which is −y2-y^2. We'll add this to the new polynomial we obtained in the previous step.

−y2+8y+5−y3−y2=−2y2+8y+5−y3-y^2 + 8y + 5 - y^3 - y^2 = -2y^2 + 8y + 5 - y^3

Step 6: Repeat the Process

We'll repeat the process of dividing the leading term of the new polynomial by the leading term of the divisor. In this case, the leading term of the new polynomial is −y3-y^3, and the leading term of the divisor is yy. We'll divide −y3-y^3 by yy to get −y2-y^2.

−y3÷y=−y2-y^3 \div y = -y^2

Step 7: Multiply the Divisor by the Quotient

Next, we'll multiply the divisor (y+1)(y+1) by the quotient −y2-y^2. This will give us a new polynomial that we'll subtract from the new polynomial we obtained in the previous step.

(y+1)×−y2=−y3−y2(y+1) \times -y^2 = -y^3 - y^2

Step 8: Subtract the Product from the New Polynomial

Now, we'll subtract the product (−y3−y2)(-y^3 - y^2) from the new polynomial (−2y2+8y+5−y3)(-2y^2 + 8y + 5 - y^3). This will give us a new polynomial that we'll use in the next step.

−2y2+8y+5−y3−(−y3−y2)=−3y2+8y+5-2y^2 + 8y + 5 - y^3 - (-y^3 - y^2) = -3y^2 + 8y + 5

Step 9: Bring Down the Next Term

We'll bring down the next term from the dividend, which is 8y8y. We'll add this to the new polynomial we obtained in the previous step.

−3y2+8y+5+8y=−3y2+16y+5-3y^2 + 8y + 5 + 8y = -3y^2 + 16y + 5

Step 10: Repeat the Process

We'll repeat the process of dividing the leading term of the new polynomial by the leading term of the divisor. In this case, the leading term of the new polynomial is −3y2-3y^2, and the leading term of the divisor is yy. We'll divide −3y2-3y^2 by yy to get −3y-3y.

−3y2÷y=−3y-3y^2 \div y = -3y

Step 11: Multiply the Divisor by the Quotient

Next, we'll multiply the divisor (y+1)(y+1) by the quotient −3y-3y. This will give us a new polynomial that we'll subtract from the new polynomial we obtained in the previous step.

(y+1)×−3y=−3y2−3y(y+1) \times -3y = -3y^2 - 3y

Step 12: Subtract the Product from the New Polynomial

Now, we'll subtract the product (−3y2−3y)(-3y^2 - 3y) from the new polynomial (−3y2+16y+5)(-3y^2 + 16y + 5). This will give us a new polynomial that we'll use in the next step.

−3y2+16y+5−(−3y2−3y)=19y+5-3y^2 + 16y + 5 - (-3y^2 - 3y) = 19y + 5

Step 13: Bring Down the Next Term

We'll bring down the next term from the dividend, which is 55. We'll add this to the new polynomial we obtained in the previous step.

19y+5+5=19y+1019y + 5 + 5 = 19y + 10

Step 14: Repeat the Process

We'll repeat the process of dividing the leading term of the new polynomial by the leading term of the divisor. In this case, the leading term of the new polynomial is 19y19y, and the leading term of the divisor is yy. We'll divide 19y19y by yy to get 1919.

19y÷y=1919y \div y = 19

Step 15: Multiply the Divisor by the Quotient

Next, we'll multiply the divisor (y+1)(y+1) by the quotient 1919. This will give us a new polynomial that we'll subtract from the new polynomial we obtained in the previous step.

(y+1)×19=19y+19(y+1) \times 19 = 19y + 19

Step 16: Subtract the Product from the New Polynomial

Now, we'll subtract the product (19y+19)(19y + 19) from the new polynomial (19y+10)(19y + 10). This will give us the remainder.

19y+10−(19y+19)=−919y + 10 - (19y + 19) = -9

Conclusion

We've successfully divided the polynomial (y4−y2+8y+5)\left(y^4-y^2+8y+5\right) by the binomial (y+1)(y+1). The quotient is y3−3y2+16y+19y^3 - 3y^2 + 16y + 19, and the remainder is −9-9. This process demonstrates the importance of polynomial division in algebra and its applications in solving equations and manipulating expressions.

Final Answer

The final answer is y3−3y2+16y+19\boxed{y^3 - 3y^2 + 16y + 19}

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Q1: What is the first step in dividing a polynomial by a binomial?

A1: The first step in dividing a polynomial by a binomial is to divide the leading term of the polynomial by the leading term of the binomial.

Q2: How do I determine the leading term of a polynomial?

A2: The leading term of a polynomial is the term with the highest degree. For example, in the polynomial y4−y2+8y+5y^4 - y^2 + 8y + 5, the leading term is y4y^4.

Q3: What is the process of polynomial division?

A3: The process of polynomial division involves dividing the leading term of the polynomial by the leading term of the binomial, multiplying the binomial by the quotient, subtracting the product from the polynomial, and repeating the process until the remainder is zero or the degree of the remainder is less than the degree of the binomial.

Q4: How do I multiply a binomial by a polynomial?

A4: To multiply a binomial by a polynomial, you can use the distributive property. For example, to multiply (y+1)(y+1) by y3y^3, you would multiply each term in the binomial by each term in the polynomial: (y+1)×y3=y4+y3(y+1) \times y^3 = y^4 + y^3.

Q5: What is the remainder in polynomial division?

A5: The remainder in polynomial division is the expression that is left over after the division process is complete. It is the result of subtracting the product of the binomial and the quotient from the polynomial.

Q6: How do I determine the degree of a polynomial?

A6: The degree of a polynomial is the highest degree of any term in the polynomial. For example, in the polynomial y4−y2+8y+5y^4 - y^2 + 8y + 5, the degree is 4.

Q7: What is the significance of polynomial division?

A7: Polynomial division is an important concept in algebra because it allows us to simplify complex expressions and solve equations. It is used in a variety of applications, including calculus, physics, and engineering.

Q8: Can I use polynomial division to divide a polynomial by a polynomial with more than two terms?

A8: Yes, you can use polynomial division to divide a polynomial by a polynomial with more than two terms. However, the process is more complex and may involve multiple steps.

Q9: How do I check my work when dividing polynomials?

A9: To check your work when dividing polynomials, you can multiply the quotient by the divisor and add the remainder to the dividend. If the result is the original polynomial, then your work is correct.

Q10: What are some common mistakes to avoid when dividing polynomials?

A10: Some common mistakes to avoid when dividing polynomials include:

  • Forgetting to divide the leading term of the polynomial by the leading term of the binomial
  • Not multiplying the binomial by the quotient correctly
  • Not subtracting the product from the polynomial correctly
  • Not repeating the process until the remainder is zero or the degree of the remainder is less than the degree of the binomial

Conclusion

Dividing polynomials by binomials is an important concept in algebra that has many practical applications. By following the steps outlined in this article and avoiding common mistakes, you can master the process of polynomial division and simplify complex expressions with ease.

Final Answer

The final answer is y3−3y2+16y+19\boxed{y^3 - 3y^2 + 16y + 19}