Math / Algebra

Question3. Expand and simplify. a) $\left(x^{3}+3 x^{2}-x+1\right)(x-2)+5$ b) $\left(2 x^{3}-4 x^{2}+x-3\right)(x+4)-7$ c) $\left(x^{3}+4 x^{2}-x+8\right)(3 x-1)+6$ d) $(x-\sqrt{2})(x+\sqrt{2})$ e) $(x-3 \sqrt{5})(x+3 \sqrt{5})$ f) $(x-1+\sqrt{3})(x-1-\sqrt{3})$

Studdy Solution

STEP 1

1. The problem involves polynomial expansion and simplification.
2. The binomial theorem and distributive property will be used for expansion.
3. The final expressions should be fully simplified.

STEP 2

1. Expand and simplify part (a).
2. Expand and simplify part (b).
3. Expand and simplify part (c).
4. Expand and simplify part (d).
5. Expand and simplify part (e).
6. Expand and simplify part (f).

STEP 3

Expand $\left(x^{3}+3 x^{2}-x+1\right)(x-2)$ using the distributive property.
$\left(x^{3}+3 x^{2}-x+1\right)(x-2) = x^3(x-2) + 3x^2(x-2) - x(x-2) + 1(x-2)$

STEP 4

Distribute each term inside the parentheses:
$x^3(x-2) = x^4 - 2x^3$ $3x^2(x-2) = 3x^3 - 6x^2$ $-x(x-2) = -x^2 + 2x$ $1(x-2) = x - 2$

STEP 5

Combine all terms from the distribution:
$(x^4 - 2x^3) + (3x^3 - 6x^2) + (-x^2 + 2x) + (x - 2)$

STEP 6

Combine like terms and add 5:
$x^4 + x^3 - 7x^2 + 3x - 2 + 5 = x^4 + x^3 - 7x^2 + 3x + 3$

STEP 7

Expand $\left(2 x^{3}-4 x^{2}+x-3\right)(x+4)$ using the distributive property.
$\left(2 x^{3}-4 x^{2}+x-3\right)(x+4) = 2x^3(x+4) - 4x^2(x+4) + x(x+4) - 3(x+4)$

STEP 8

Distribute each term inside the parentheses:
$2x^3(x+4) = 2x^4 + 8x^3$ $-4x^2(x+4) = -4x^3 - 16x^2$ $x(x+4) = x^2 + 4x$ $-3(x+4) = -3x - 12$

STEP 9

Combine all terms from the distribution:
$(2x^4 + 8x^3) + (-4x^3 - 16x^2) + (x^2 + 4x) + (-3x - 12)$

STEP 10

Combine like terms and subtract 7:
$2x^4 + 4x^3 - 15x^2 + x - 12 - 7 = 2x^4 + 4x^3 - 15x^2 + x - 19$

STEP 11

Expand $\left(x^{3}+4 x^{2}-x+8\right)(3 x-1)$ using the distributive property.
$\left(x^{3}+4 x^{2}-x+8\right)(3 x-1) = x^3(3x-1) + 4x^2(3x-1) - x(3x-1) + 8(3x-1)$

STEP 12

Distribute each term inside the parentheses:
$x^3(3x-1) = 3x^4 - x^3$ $4x^2(3x-1) = 12x^3 - 4x^2$ $-x(3x-1) = -3x^2 + x$ $8(3x-1) = 24x - 8$

STEP 13

Combine all terms from the distribution:
$(3x^4 - x^3) + (12x^3 - 4x^2) + (-3x^2 + x) + (24x - 8)$

STEP 14

Combine like terms and add 6:
$3x^4 + 11x^3 - 7x^2 + 25x - 8 + 6 = 3x^4 + 11x^3 - 7x^2 + 25x - 2$

STEP 15

Expand $(x-\sqrt{2})(x+\sqrt{2})$ using the difference of squares formula:
$(x-\sqrt{2})(x+\sqrt{2}) = x^2 - (\sqrt{2})^2$

STEP 16

Simplify the expression:
$x^2 - 2$

STEP 17

Expand $(x-3\sqrt{5})(x+3\sqrt{5})$ using the difference of squares formula:
$(x-3\sqrt{5})(x+3\sqrt{5}) = x^2 - (3\sqrt{5})^2$

STEP 18

Simplify the expression:
$x^2 - 45$

STEP 19

Expand $(x-1+\sqrt{3})(x-1-\sqrt{3})$ using the distributive property.
$(x-1+\sqrt{3})(x-1-\sqrt{3}) = (x-1)^2 - (\sqrt{3})^2$

STEP 20

First, expand $(x-1)^2$ :
$(x-1)^2 = x^2 - 2x + 1$

STEP 21

Combine the terms:
$(x-1)^2 - (\sqrt{3})^2 = x^2 - 2x + 1 - 3$

STEP 22

Simplify the expression:
$x^2 - 2x - 2$
Solutions: a) $x^4 + x^3 - 7x^2 + 3x + 3$ b) $2x^4 + 4x^3 - 15x^2 + x - 19$ c) $3x^4 + 11x^3 - 7x^2 + 25x - 2$ d) $x^2 - 2$ e) $x^2 - 45$ f) $x^2 - 2x - 2$

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Studdy Solution

STEP 1

1. The problem involves polynomial expansion and simplification.2. The binomial theorem and distributive property will be used for expansion.3. The final expressions should be fully simplified.

STEP 2

1. Expand and simplify part (a).2. Expand and simplify part (b).3. Expand and simplify part (c).4. Expand and simplify part (d).5. Expand and simplify part (e).6. Expand and simplify part (f).

STEP 3

STEP 4

Distribute each term inside the parentheses:x3(x−2)=x4−2x3 x^3(x-2) = x^4 - 2x^3 x3(x−2)=x4−2x3 3x2(x−2)=3x3−6x2 3x^2(x-2) = 3x^3 - 6x^2 3x2(x−2)=3x3−6x2 −x(x−2)=−x2+2x -x(x-2) = -x^2 + 2x −x(x−2)=−x2+2x 1(x−2)=x−2 1(x-2) = x - 2 1(x−2)=x−2

STEP 5

Combine all terms from the distribution:(x4−2x3)+(3x3−6x2)+(−x2+2x)+(x−2) (x^4 - 2x^3) + (3x^3 - 6x^2) + (-x^2 + 2x) + (x - 2) (x4−2x3)+(3x3−6x2)+(−x2+2x)+(x−2)

STEP 6

Combine like terms and add 5:x4+x3−7x2+3x−2+5=x4+x3−7x2+3x+3 x^4 + x^3 - 7x^2 + 3x - 2 + 5 = x^4 + x^3 - 7x^2 + 3x + 3 x4+x3−7x2+3x−2+5=x4+x3−7x2+3x+3

STEP 7

STEP 8

Distribute each term inside the parentheses:2x3(x+4)=2x4+8x3 2x^3(x+4) = 2x^4 + 8x^3 2x3(x+4)=2x4+8x3 −4x2(x+4)=−4x3−16x2 -4x^2(x+4) = -4x^3 - 16x^2 −4x2(x+4)=−4x3−16x2 x(x+4)=x2+4x x(x+4) = x^2 + 4x x(x+4)=x2+4x −3(x+4)=−3x−12 -3(x+4) = -3x - 12 −3(x+4)=−3x−12

STEP 9

Combine all terms from the distribution:(2x4+8x3)+(−4x3−16x2)+(x2+4x)+(−3x−12) (2x^4 + 8x^3) + (-4x^3 - 16x^2) + (x^2 + 4x) + (-3x - 12) (2x4+8x3)+(−4x3−16x2)+(x2+4x)+(−3x−12)

STEP 10

Combine like terms and subtract 7:2x4+4x3−15x2+x−12−7=2x4+4x3−15x2+x−19 2x^4 + 4x^3 - 15x^2 + x - 12 - 7 = 2x^4 + 4x^3 - 15x^2 + x - 19 2x4+4x3−15x2+x−12−7=2x4+4x3−15x2+x−19

STEP 11

STEP 12

Distribute each term inside the parentheses:x3(3x−1)=3x4−x3 x^3(3x-1) = 3x^4 - x^3 x3(3x−1)=3x4−x3 4x2(3x−1)=12x3−4x2 4x^2(3x-1) = 12x^3 - 4x^2 4x2(3x−1)=12x3−4x2 −x(3x−1)=−3x2+x -x(3x-1) = -3x^2 + x −x(3x−1)=−3x2+x 8(3x−1)=24x−8 8(3x-1) = 24x - 8 8(3x−1)=24x−8

STEP 13

Combine all terms from the distribution:(3x4−x3)+(12x3−4x2)+(−3x2+x)+(24x−8) (3x^4 - x^3) + (12x^3 - 4x^2) + (-3x^2 + x) + (24x - 8) (3x4−x3)+(12x3−4x2)+(−3x2+x)+(24x−8)

STEP 14

Combine like terms and add 6:3x4+11x3−7x2+25x−8+6=3x4+11x3−7x2+25x−2 3x^4 + 11x^3 - 7x^2 + 25x - 8 + 6 = 3x^4 + 11x^3 - 7x^2 + 25x - 2 3x4+11x3−7x2+25x−8+6=3x4+11x3−7x2+25x−2

STEP 15

Expand (x−2)(x+2)(x-\sqrt{2})(x+\sqrt{2})(x−2​)(x+2​) using the difference of squares formula:(x−2)(x+2)=x2−(2)2 (x-\sqrt{2})(x+\sqrt{2}) = x^2 - (\sqrt{2})^2 (x−2​)(x+2​)=x2−(2​)2

STEP 16

Simplify the expression:x2−2 x^2 - 2 x2−2

STEP 17

Expand (x−35)(x+35)(x-3\sqrt{5})(x+3\sqrt{5})(x−35​)(x+35​) using the difference of squares formula:(x−35)(x+35)=x2−(35)2 (x-3\sqrt{5})(x+3\sqrt{5}) = x^2 - (3\sqrt{5})^2 (x−35​)(x+35​)=x2−(35​)2

STEP 18

Simplify the expression:x2−45 x^2 - 45 x2−45

STEP 19

Expand (x−1+3)(x−1−3)(x-1+\sqrt{3})(x-1-\sqrt{3})(x−1+3​)(x−1−3​) using the distributive property.(x−1+3)(x−1−3)=(x−1)2−(3)2 (x-1+\sqrt{3})(x-1-\sqrt{3}) = (x-1)^2 - (\sqrt{3})^2 (x−1+3​)(x−1−3​)=(x−1)2−(3​)2

STEP 20

First, expand (x−1)2(x-1)^2(x−1)2:(x−1)2=x2−2x+1 (x-1)^2 = x^2 - 2x + 1 (x−1)2=x2−2x+1

STEP 21

Combine the terms:(x−1)2−(3)2=x2−2x+1−3 (x-1)^2 - (\sqrt{3})^2 = x^2 - 2x + 1 - 3 (x−1)2−(3​)2=x2−2x+1−3

STEP 22

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1. The problem involves polynomial expansion and simplification.
2. The binomial theorem and distributive property will be used for expansion.
3. The final expressions should be fully simplified.

1. Expand and simplify part (a).
2. Expand and simplify part (b).
3. Expand and simplify part (c).
4. Expand and simplify part (d).
5. Expand and simplify part (e).
6. Expand and simplify part (f).

Distribute each term inside the parentheses:
$x^3(x-2) = x^4 - 2x^3$ $3x^2(x-2) = 3x^3 - 6x^2$ $-x(x-2) = -x^2 + 2x$ $1(x-2) = x - 2$

Combine all terms from the distribution:
$(x^4 - 2x^3) + (3x^3 - 6x^2) + (-x^2 + 2x) + (x - 2)$

Combine like terms and add 5:
$x^4 + x^3 - 7x^2 + 3x - 2 + 5 = x^4 + x^3 - 7x^2 + 3x + 3$

Distribute each term inside the parentheses:
$2x^3(x+4) = 2x^4 + 8x^3$ $-4x^2(x+4) = -4x^3 - 16x^2$ $x(x+4) = x^2 + 4x$ $-3(x+4) = -3x - 12$

Combine all terms from the distribution:
$(2x^4 + 8x^3) + (-4x^3 - 16x^2) + (x^2 + 4x) + (-3x - 12)$

Combine like terms and subtract 7:
$2x^4 + 4x^3 - 15x^2 + x - 12 - 7 = 2x^4 + 4x^3 - 15x^2 + x - 19$

Distribute each term inside the parentheses:
$x^3(3x-1) = 3x^4 - x^3$ $4x^2(3x-1) = 12x^3 - 4x^2$ $-x(3x-1) = -3x^2 + x$ $8(3x-1) = 24x - 8$

Combine all terms from the distribution:
$(3x^4 - x^3) + (12x^3 - 4x^2) + (-3x^2 + x) + (24x - 8)$

Combine like terms and add 6:
$3x^4 + 11x^3 - 7x^2 + 25x - 8 + 6 = 3x^4 + 11x^3 - 7x^2 + 25x - 2$

Expand $(x-\sqrt{2})(x+\sqrt{2})$ using the difference of squares formula:
$(x-\sqrt{2})(x+\sqrt{2}) = x^2 - (\sqrt{2})^2$

Simplify the expression:
$x^2 - 2$

Expand $(x-3\sqrt{5})(x+3\sqrt{5})$ using the difference of squares formula:
$(x-3\sqrt{5})(x+3\sqrt{5}) = x^2 - (3\sqrt{5})^2$

Simplify the expression:
$x^2 - 45$

Expand $(x-1+\sqrt{3})(x-1-\sqrt{3})$ using the distributive property.
$(x-1+\sqrt{3})(x-1-\sqrt{3}) = (x-1)^2 - (\sqrt{3})^2$

First, expand $(x-1)^2$ :
$(x-1)^2 = x^2 - 2x + 1$

Combine the terms:
$(x-1)^2 - (\sqrt{3})^2 = x^2 - 2x + 1 - 3$