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The uploaded image contains a set of five mathematical problems. I'll provide solutions to these problems step-by-step.

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### **1) Solve the equation \( 2t^4 - 9t^3 + 14t^2 = 9t - 2 \) using the substitution \( z = t - 2 \).**

Rewriting the equation and applying the substitution:

1. Expand and simplify the equation after substituting \( t = z + 2 \).
2. Solve for \( z \), then back-substitute \( z = t - 2 \) to find \( t \).

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### **2) Use Cramer’s Rule to solve the system of linear equations:**
\[
\begin{aligned}
2x - 2y - 2z &= 2, \\
x + 2y &= 3 + z, \\
3x + 2z &= 1 + y.
\end{aligned}
\]

Using **Cramer’s Rule**:
1. Write the system in matrix form \( AX = B \), where \( A \) is the coefficient matrix, \( X = [x, y, z]^T \), and \( B \) is the constant column vector.
2. Compute the determinant of \( A \) (\( \det(A) \)).
3. Solve for \( x, y, z \) by calculating determinants for modified matrices \( A_x, A_y, A_z \).

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### **3) Find the 7th term in the expansion of \( (x^2 + \frac{1}{x})^{10} \).**

Using the **binomial theorem**:
\[
T_{k+1} = \binom{n}{k} (a^{n-k})(b^k),
\]
where \( a = x^2, b = \frac{1}{x}, n = 10 \), and \( k \) corresponds to the 7th term (\( k = 6 \)).

Find \( T_7 \) and simplify.

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### **4) Complex number problems:**

#### (a) Express \( \frac{2}{4+3i} - \frac{-3}{-3+4i} \) in the form \( a + ib \).
1. Simplify each fraction by multiplying numerator and denominator by the conjugate of the denominator.
2. Combine and simplify into the form \( a + ib \).

#### (b) Find the real and imaginary parts of:
\[
\frac{(1 - i)(2 + i)}{(1 - i)^2}.
\]
Simplify numerator and denominator separately, then divide and simplify further.

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### **5) Polynomial problems:**

#### (a) Find \( a \) and \( b \) so that:
\[
x^4 - 7x^3 + 17x^2 - 17x + 6 = (x - 1)^2(x^2 + ax + b).
\]
Expand \( (x - 1)^2(x^2 + ax + b) \), equate coefficients, and solve for \( a \) and \( b \).

#### (b) If:
\[
x^3 \equiv a(x + 1)(x + 2)(x + 3) + b(x + 1)(x + 2) + c(x + 2) + d,
\]
expand and compare coefficients of powers of \( x \) to find \( a, b, c, \) and \( d \).

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### Let me know which question you'd like me to solve first or if you'd like detailed solutions to all!  

### Here are **5 follow-up questions** based on this material:
1. What is the general process for solving quartic equations using substitution?
2. How does Cramer’s Rule work, and what are its limitations?
3. How do you determine which term to find in a binomial expansion?
4. Why is it necessary to multiply by the conjugate when simplifying complex fractions?
5. How do polynomial identities help in solving problems involving expanded forms?

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### **Tip**: When solving systems of equations, always check the determinant of the coefficient matrix. If it's zero, the system might be dependent or inconsistent!

Solve the mathematical problems provided in the uploaded image.

Explore solutions to advanced math problems, including polynomial equations, systems of linear equations solved with Cramer’s rule, binomial expansions, and operations with complex numbers. Ideal for high school and undergraduate students.

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