CameraIcon

SearchIcon

MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Multiplication of a Polynomial and Monomial

Expand a + b ...

Question

Expand ${(a + b)^{4} + (a - b)^{4}}$ and use it to ealuate
$(x^{2} + \sqrt{1 - x^{2}})^{4} + (x^{2} - \sqrt{1 - x^{2}})^{4}$

Open in App

Solution

We have
$(a + b)^{4} + (a - b)^{4} = [^{4} C_{0} a^{4} +^{4} C_{1} a^{3} b +^{4} C_{2} a^{2} b^{2} +^{6} C_{3} a b^{3} +^{4} C_{4} b^{4}] + [^{4} C_{0} a^{4} -^{4} C_{1} a^{3} b +^{4} C_{2} a^{2} b^{2} -^{4} c_{3} a b^{3} +^{4} C_{4} b^{4}] = 2 [^{4} C_{0} a^{4} +^{4} C_{2} a^{2} b^{2} +^{4} C_{4} b^{4}] Putting \sqrt{1 - x^{2}} = y, we get (x^{2} + \sqrt{1 - x^{2}})^{4} + (x^{2} - \sqrt{1 - x^{2}})^{4} (x^{2} + y)^{4} + (x^{2} - y)^{4} = 2 [^{4} C_{0} (x^{2})^{4} +^{4} C_{2} (x^{2})^{2} y^{2} +^{4} C_{4} y^{4}] = 2 [x^{6} + 6 x^{4} y^{2} + y^{4}] = 2 [x^{8} + 6 x^{4} (1 - x^{2}) + (1 - x^{2})^{2}] = 2 [x^{8} + (6 x^{4} - 6 x^{6}) + (1 - 2 x^{2} + x^{2})] = 2 [x^{8} - 6 x^{6} + 7 x^{4} - 2 x^{2} + 1] = 2 x^{8} - 12 x^{6} + 14 x^{4} - 4 x^{2} + 2$

Suggest Corrections

thumbs-up

3

Similar questions

{(x^{2} - \sqrt{1 - x^{2}})}^{4} + {(x^{2} + \sqrt{1 - x^{2}})}^{4}

Q. The coefficient of

x^{24}

{(1 + x^{2})}^{12} \cdot (1 + x^{12}) \cdot (1 + x^{24})

If X = 1 - a^{2} and Y = 1 - b^{2}, then what is X^{2} + Y^{2} ?

{(a + b)}^{4}

\int_{0}^{\infty} \frac{d x}{(1 + x^{2})^{4}}

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

Multiplication of a Polynomial and Monomial

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Multiplication of a Polynomial and Monomial

Standard VI Mathematics

Join BYJU'S Learning Program

CrossIcon