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To factorise the expression [tex]\( a^4 - 13a^2b^2 + 4b^4 \)[/tex], we can follow these steps:
1. Identify the polynomial:
The given polynomial is [tex]\( a^4 - 13a^2b^2 + 4b^4 \)[/tex].
2. Look for a factorisation pattern:
Notice that the polynomial can potentially be factored into the product of two quadratic expressions. We will look for factors of the form:
[tex]\[ (a^2 + m_1ab + n_1b^2)(a^2 + m_2ab + n_2b^2) \][/tex]
where [tex]\(m_1\)[/tex], [tex]\(m_2\)[/tex], [tex]\(n_1\)[/tex], and [tex]\(n_2\)[/tex] are constants we need to determine.
3. Expand the candidate factors:
When we expand [tex]\((a^2 + m_1ab + n_1b^2)(a^2 + m_2ab + n_2b^2)\)[/tex], we should get:
[tex]\[ a^4 + (m_1 + m_2)a^3b + (n_1 + n_2 + m_1m_2)a^2b^2 + (m_1n_2 + m_2n_1)ab^3 + n_1n_2b^4 \][/tex]
For the given polynomial [tex]\(a^4 - 13a^2b^2 + 4b^4\)[/tex], we observe that the coefficients of [tex]\(a^3b\)[/tex] and [tex]\(ab^3\)[/tex] terms must be zero. Therefore, [tex]\(m_1 + m_2 = 0\)[/tex] and [tex]\(m_1n_2 + m_2n_1 = 0\)[/tex].
4. Determine specific values for [tex]\(m_1\)[/tex], [tex]\(m_2\)[/tex], [tex]\(n_1\)[/tex], and [tex]\(n_2\)[/tex]:
From [tex]\(m_1 + m_2 = 0\)[/tex], we can set [tex]\(m_2 = -m_1\)[/tex]. We also need [tex]\(n_1 + n_2 + m_1m_2 = -13\)[/tex] and [tex]\(n_1n_2 = 4\)[/tex]. Let's solve these by choosing [tex]\(m_1 = 3\)[/tex] and [tex]\(m_2 = -3\)[/tex] (since [tex]\(3 \cdot (-3) = -9\)[/tex]), and check the [tex]\(n_1\)[/tex] and [tex]\(n_2\)[/tex] values later.
Then, with [tex]\( n_1 + n_2 = -13 + 9 = -4 \)[/tex] and [tex]\( n_1n_2 = 4 \)[/tex], we solve the quadratic equation [tex]\( t^2 + 4t + 4 = 0 \)[/tex]:
[tex]\[ t = -2 \quad (\text{repeated root}) \][/tex]
Therefore, it satisfies [tex]\(n_1 = -2\)[/tex] and [tex]\(n_2 = -2\)[/tex].
5. Check the factorisation:
Hence, our factors become:
[tex]\[ (a^2 - 3ab - 2b^2) \quad \text{and} \quad (a^2 + 3ab - 2b^2) \][/tex]
6. Writing the factorisation:
Thus, the polynomial [tex]\( a^4 - 13a^2b^2 + 4b^4 \)[/tex] factors as:
[tex]\[ (a^2 - 3ab - 2b^2)(a^2 + 3ab - 2b^2) \][/tex]
So, the factorised form is:
[tex]\[ a^4 - 13a^2b^2 + 4b^4 = (a^2 - 3ab - 2b^2)(a^2 + 3ab - 2b^2) \][/tex]
1. Identify the polynomial:
The given polynomial is [tex]\( a^4 - 13a^2b^2 + 4b^4 \)[/tex].
2. Look for a factorisation pattern:
Notice that the polynomial can potentially be factored into the product of two quadratic expressions. We will look for factors of the form:
[tex]\[ (a^2 + m_1ab + n_1b^2)(a^2 + m_2ab + n_2b^2) \][/tex]
where [tex]\(m_1\)[/tex], [tex]\(m_2\)[/tex], [tex]\(n_1\)[/tex], and [tex]\(n_2\)[/tex] are constants we need to determine.
3. Expand the candidate factors:
When we expand [tex]\((a^2 + m_1ab + n_1b^2)(a^2 + m_2ab + n_2b^2)\)[/tex], we should get:
[tex]\[ a^4 + (m_1 + m_2)a^3b + (n_1 + n_2 + m_1m_2)a^2b^2 + (m_1n_2 + m_2n_1)ab^3 + n_1n_2b^4 \][/tex]
For the given polynomial [tex]\(a^4 - 13a^2b^2 + 4b^4\)[/tex], we observe that the coefficients of [tex]\(a^3b\)[/tex] and [tex]\(ab^3\)[/tex] terms must be zero. Therefore, [tex]\(m_1 + m_2 = 0\)[/tex] and [tex]\(m_1n_2 + m_2n_1 = 0\)[/tex].
4. Determine specific values for [tex]\(m_1\)[/tex], [tex]\(m_2\)[/tex], [tex]\(n_1\)[/tex], and [tex]\(n_2\)[/tex]:
From [tex]\(m_1 + m_2 = 0\)[/tex], we can set [tex]\(m_2 = -m_1\)[/tex]. We also need [tex]\(n_1 + n_2 + m_1m_2 = -13\)[/tex] and [tex]\(n_1n_2 = 4\)[/tex]. Let's solve these by choosing [tex]\(m_1 = 3\)[/tex] and [tex]\(m_2 = -3\)[/tex] (since [tex]\(3 \cdot (-3) = -9\)[/tex]), and check the [tex]\(n_1\)[/tex] and [tex]\(n_2\)[/tex] values later.
Then, with [tex]\( n_1 + n_2 = -13 + 9 = -4 \)[/tex] and [tex]\( n_1n_2 = 4 \)[/tex], we solve the quadratic equation [tex]\( t^2 + 4t + 4 = 0 \)[/tex]:
[tex]\[ t = -2 \quad (\text{repeated root}) \][/tex]
Therefore, it satisfies [tex]\(n_1 = -2\)[/tex] and [tex]\(n_2 = -2\)[/tex].
5. Check the factorisation:
Hence, our factors become:
[tex]\[ (a^2 - 3ab - 2b^2) \quad \text{and} \quad (a^2 + 3ab - 2b^2) \][/tex]
6. Writing the factorisation:
Thus, the polynomial [tex]\( a^4 - 13a^2b^2 + 4b^4 \)[/tex] factors as:
[tex]\[ (a^2 - 3ab - 2b^2)(a^2 + 3ab - 2b^2) \][/tex]
So, the factorised form is:
[tex]\[ a^4 - 13a^2b^2 + 4b^4 = (a^2 - 3ab - 2b^2)(a^2 + 3ab - 2b^2) \][/tex]
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