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Answer :
Answer:
Explanation:
We are given the polynomial expression:
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
2x
4
+5x
3
−20x
2
−20x+48
To simplify this polynomial, we need to look for common factors or possible factorizations. However, this expression does not have any obvious common factors among all terms, so let's attempt to factor it by grouping or by using synthetic division.
Step 1: Check for possible rational roots using the Rational Root Theorem
The Rational Root Theorem suggests that any rational root of the polynomial should be of the form
�
�
q
p
, where:
�
p is a factor of the constant term (48).
�
q is a factor of the leading coefficient (2).
The constant term is
48
48, and its factors are:
±
1
,
±
2
,
±
3
,
±
4
,
±
6
,
±
8
,
±
12
,
±
16
,
±
24
,
±
48
±1,±2,±3,±4,±6,±8,±12,±16,±24,±48
The leading coefficient is
2
2, and its factors are:
±
1
,
±
2
±1,±2
Thus, the possible rational roots are:
±
1
,
±
2
,
±
1
2
,
±
3
,
±
4
,
±
3
2
,
±
6
,
±
8
,
±
4
2
,
±
12
,
±
16
,
±
12
2
,
±
24
,
±
48
±1,±2,±
2
1
,±3,±4,±
2
3
,±6,±8,±
2
4
,±12,±16,±
2
12
,±24,±48
Step 2: Try possible roots using synthetic division
Let's try testing some simple possible roots such as
�
=
2
x=2.
Testing
�
=
2
x=2:
Perform synthetic division on
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
2x
4
+5x
3
−20x
2
−20x+48 by
�
−
2
x−2:
2
2
5
−
20
−
20
48
4
18
−
4
−
48
2
9
−
2
−
24
0
2
2
2
5
4
9
−20
18
−2
−20
−4
−24
48
−48
0
Since the remainder is 0,
�
=
2
x=2 is a root. Therefore, we can factor out
(
�
−
2
)
(x−2).
Step 3: Factor the quotient polynomial
The quotient from the synthetic division is:
2
�
3
+
9
�
2
−
2
�
−
24
2x
3
+9x
2
−2x−24
Now, let's factor
2
�
3
+
9
�
2
−
2
�
−
24
2x
3
+9x
2
−2x−24. We can try grouping:
(
2
�
3
+
9
�
2
)
−
(
2
�
+
24
)
(2x
3
+9x
2
)−(2x+24)
Factor each group:
�
2
(
2
�
+
9
)
−
2
(
2
�
+
9
)
x
2
(2x+9)−2(2x+9)
Now, factor out the common binomial factor
(
2
�
+
9
)
(2x+9):
(
�
2
−
2
)
(
2
�
+
9
)
(x
2
−2)(2x+9)
Step 4: Combine all factors
We have factored the original polynomial as:
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
=
(
�
−
2
)
(
�
2
−
2
)
(
2
�
+
9
)
2x
4
+5x
3
−20x
2
−20x+48=(x−2)(x
2
−2)(2x+9)
This is the simplified and fully factored form of the given polynomial.
Final Answer:
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
=
(
�
−
2
)
(
�
2
−
2
)
(
2
�
+
9
)
2x
4
+5x
3
−20x
2
−20x+48=(x−2)(x
2
−2)(2x+9)
We are given the polynomial expression:
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
2x
4
+5x
3
−20x
2
−20x+48
To simplify this polynomial, we need to look for common factors or possible factorizations. However, this expression does not have any obvious common factors among all terms, so let's attempt to factor it by grouping or by using synthetic division.
Step 1: Check for possible rational roots using the Rational Root Theorem
The Rational Root Theorem suggests that any rational root of the polynomial should be of the form
�
�
q
p
, where:
�
p is a factor of the constant term (48).
�
q is a factor of the leading coefficient (2).
The constant term is
48
48, and its factors are:
±
1
,
±
2
,
±
3
,
±
4
,
±
6
,
±
8
,
±
12
,
±
16
,
±
24
,
±
48
±1,±2,±3,±4,±6,±8,±12,±16,±24,±48
The leading coefficient is
2
2, and its factors are:
±
1
,
±
2
±1,±2
Thus, the possible rational roots are:
±
1
,
±
2
,
±
1
2
,
±
3
,
±
4
,
±
3
2
,
±
6
,
±
8
,
±
4
2
,
±
12
,
±
16
,
±
12
2
,
±
24
,
±
48
±1,±2,±
2
1
,±3,±4,±
2
3
,±6,±8,±
2
4
,±12,±16,±
2
12
,±24,±48
Step 2: Try possible roots using synthetic division
Let's try testing some simple possible roots such as
�
=
2
x=2.
Testing
�
=
2
x=2:
Perform synthetic division on
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
2x
4
+5x
3
−20x
2
−20x+48 by
�
−
2
x−2:
2
2
5
−
20
−
20
48
4
18
−
4
−
48
2
9
−
2
−
24
0
2
2
2
5
4
9
−20
18
−2
−20
−4
−24
48
−48
0
Since the remainder is 0,
�
=
2
x=2 is a root. Therefore, we can factor out
(
�
−
2
)
(x−2).
Step 3: Factor the quotient polynomial
The quotient from the synthetic division is:
2
�
3
+
9
�
2
−
2
�
−
24
2x
3
+9x
2
−2x−24
Now, let's factor
2
�
3
+
9
�
2
−
2
�
−
24
2x
3
+9x
2
−2x−24. We can try grouping:
(
2
�
3
+
9
�
2
)
−
(
2
�
+
24
)
(2x
3
+9x
2
)−(2x+24)
Factor each group:
�
2
(
2
�
+
9
)
−
2
(
2
�
+
9
)
x
2
(2x+9)−2(2x+9)
Now, factor out the common binomial factor
(
2
�
+
9
)
(2x+9):
(
�
2
−
2
)
(
2
�
+
9
)
(x
2
−2)(2x+9)
Step 4: Combine all factors
We have factored the original polynomial as:
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
=
(
�
−
2
)
(
�
2
−
2
)
(
2
�
+
9
)
2x
4
+5x
3
−20x
2
−20x+48=(x−2)(x
2
−2)(2x+9)
This is the simplified and fully factored form of the given polynomial.
Final Answer:
2
�
4
+
5
�
3
−
20
�
2
−
20
�
+
48
=
(
�
−
2
)
(
�
2
−
2
(2+9)2x 4 +5x 3 −20x 2 −20x+48=(x−2)(x 2 −2)(2x+9)
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Rewritten by : Barada
To solve the polynomial [tex]\(2x^4 + 5x^3 - 20x^2 - 20x + 48\)[/tex] by factoring, let's go through a step-by-step process:
1. Identify the Polynomial: The expression we need to factor is [tex]\(2x^4 + 5x^3 - 20x^2 - 20x + 48\)[/tex].
2. Look for Common Factors: Check if there's a common factor for all the terms. In this case, there isn't a common factor aside from 1.
3. Apply Polynomial Factoring Techniques: We will approach the polynomial by attempting to factor it into simpler expressions.
4. Use Trial and Error with Rational Root Theorem: Finding any potential rational roots can help break down the polynomial into a product of lower-degree polynomials. Using methods or specific calculations, we could find these roots.
5. Breaking Down into Factors:
- The polynomial can be factored as [tex]\((x - 2)(x + 2)(x + 4)(2x - 3)\)[/tex].
6. Verify the Factors:
- Each of these factors corresponds to a root of the polynomial.
- If we expand these factors, they will give the original polynomial expression.
By following these steps, the polynomial [tex]\(2x^4 + 5x^3 - 20x^2 - 20x + 48\)[/tex] can be factorized into [tex]\((x - 2)(x + 2)(x + 4)(2x - 3)\)[/tex]. This completes the solution.
1. Identify the Polynomial: The expression we need to factor is [tex]\(2x^4 + 5x^3 - 20x^2 - 20x + 48\)[/tex].
2. Look for Common Factors: Check if there's a common factor for all the terms. In this case, there isn't a common factor aside from 1.
3. Apply Polynomial Factoring Techniques: We will approach the polynomial by attempting to factor it into simpler expressions.
4. Use Trial and Error with Rational Root Theorem: Finding any potential rational roots can help break down the polynomial into a product of lower-degree polynomials. Using methods or specific calculations, we could find these roots.
5. Breaking Down into Factors:
- The polynomial can be factored as [tex]\((x - 2)(x + 2)(x + 4)(2x - 3)\)[/tex].
6. Verify the Factors:
- Each of these factors corresponds to a root of the polynomial.
- If we expand these factors, they will give the original polynomial expression.
By following these steps, the polynomial [tex]\(2x^4 + 5x^3 - 20x^2 - 20x + 48\)[/tex] can be factorized into [tex]\((x - 2)(x + 2)(x + 4)(2x - 3)\)[/tex]. This completes the solution.
